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Theory of the Motion of the Moon; containing a New Calculation of the Expres- 
sions for the Coordinates of the Moon in terms of the Time. By Ernest 
W. Bkown, M.A., Sc.D. 



PART I. CHAPTERS I. -IV 



INTRODUCTION. 

The formation of numerical expressions, deduced as a consequence of the 
Newtonian laws of motion and gravitation, which shall represent the position 
of the Moon at any time, may be roughly divided into three stages. As a 
first step we consider each of the three bodies — the Sun, the Earth, and the 
Moon — as a sphere of mass equal to its actual mass and arranged in concentric 
spherical layers of equal density. The Earth (or the centre of mass of the Earth 
and Moon) is supposed to move round the Sun in a certain ideal elliptic orbit, 
and all disturbances of this orbit and of the Moon from any other source than 
the ideal Sun and Earth are neglected. This first stage constitutes nearly 
the whole of the labour of solving the problem of three bodies as far as the 
particular configuration of the Sun-Earth-Moon system is concerned. When 
this is done we proceed to the second step, which involves the determination 
of the effects due to the difference between the actual and ideal motions of 
the Earth and Sun, to the influences exerted by the other bodies of the solar 
system, and to the differences between the real and ideal arrangements of the 
masses of the bodies. The calculations so far may, theoretically at least, be 
made without any knowledge of the configuration of the system at any given 



mm 



40 Mr. Ernest "W. Brown, Theory of the 

time or times beyond a general idea of the order of magnitude of certain of 
the constants involved. The third and final stage consists in a determination 
by observation of the various constants which have entered into the theory, 
and the substitution of their values so as to obtain numerical expressions for 
the coordinates in terms of the time. 

In actual practice these lines of division cannot be satisfactorily kept, 
partly from the length and complexity of the calculations necessary to obtain 
algebraical expressions sufficiently complete, and partly from a similar diffi- 
culty in comparing large numbers of observations with the results of the 
theory for the purpose of obtaining the values of the constants involved. 
Certain of the latter, however, particularly those most frequently occurring, 
namely, the mean motions of the Sun (or Earth) and Moon, can be obtained, , 
with very little knowledge of the theory, with sufficient accuracy to enable 
us to use their numerical values at the outset, and so to save a large part of 
the labour. The first and second steps may be separated without much 
difficulty, and it is the first which forms the chief object of this Memoir. 

Of the methods which have been devised to solve the first part of the 

problem, those of Hansen and Delaunay must take the first place, not 

because they were intrinsically the best adapted to the purpose, but because 

in the hands of their authors they were actually carried out with a degree of 

detail greater than any other. It is not necessary to discuss here their 

respective merits ; it is sufficient to recall the fact that Hansen's was entirely 

numerical, and was made the basis of the tables of the Moon used at the 

present time, although the latter were published forty years ago, while 

Delaunay's was, owing to the method used, entirely algebraical. Further, 

Hansen embraced the whole problem in his theory ; Delaunay only lived 

long enough to complete the first step ; his work, however, has been carried 

on by other writers, and will attain its completion in the tables now in 

process of formation. The earlier method of Laplace, adopted also by 

Damoiseau and Plana, and that of Lubbock and de Pontecoulant, although 

perhaps unsuitable to obtain the accuracy required at the present day, 

attained results which were at the time as much in advance of those previously 

published as those of Hansen and Delaunay were over all earlier theories. 

But if we look only to suggestions of methods of treatment, quite apart 
from the extent to which they were actually carried out, the most fruitful 



Motion of the Moon. 



4i 



contributions to the first part of the problem were undoubtedly those of 
Euler, who preceded all the writers just mentioned. His two treatises of 
1753 and 1772 contain three distinct methods ; and it is not a little remark- 
able that the theories of Hansen and Delaunay may be said to be ultimately 
based on two of them, while the third forms the foundation of the method 
developed below. Amongst the points of correspondence between the first 
theory of Euler and the theory of Hansen, it is sufficient to note the manner 
in which the true longitude and the radius vector of the Moon are expressed. 
Hansen, however, covered up most of the traces of any such connection by 
his peculiar method of using a so-called " variable time " and by developing 
his formula} with the aid of the method of variation of arbitrary constants. 
His formula} can be otherwise obtained, as Hansen himself and others have 
shown ; even if it were not so, we owe the application of the arbitrary 
constants method to celestial mechanics in the first place to Euler. This 
latter method, which is contained in the appendix to Euler's volume of 1753, 
was rendered a practical one under the masterly treatment of Delaunay ; but 
it is doubtful if anyone but the originator of the method would have had the 
courage to undertake the laborious calculations necessary to bring the work- 
to a successful conclusion. 

The theory of Euler, published in 1772, is of particular interest here, 
since it suggested the method used below. It is based on the use of rect- 
angular axes, of which two move in their own plane with constant angular 
velocity, and on a division of the inequalities into classes according to the 
composition of their coefficients. This work of Euler seems to have 
received but little attention, apart from the practical results obtained ; no 
attempt was made to develop his method further. It was reserved for Dr. 
Gr. W. Hill, over a hundred years later, to take the next step by so altering 
the forms of the equations, while preserving the original ideas on which they 
were constructed, that they might be made available for accurate calculation. 
His two papers, " Researches in the Lunar Theory," published in the first 
volume of the American Journal of Mathematics, and " On the Part of the 
Motion of the Lunar Perigee which is a Function of the Mean Motions of 
the Sun and Moon," published separately, and also in the eighth volume of 
the Acta Mathematica, showed what the method was capable of effecting, and 
opened out a new region for practical calculations and theoretical researches. 

Royal Astron. Soc, Vol. LIII. h 



4 2 Mr. Ernest W. Brown, Theory of the 

" Dans cetts oeuvre," says M. Poist:are in the preface to the first volume of 
his Mecanique Celeste, " il est parmis d'apercevoir le germe de la plupart des 
progres que la science a faits depuis." 

In this connection the work of Adams must be mentioned. Starting 
from an entirely different point of view, that of effecting an accurate 
determination of the mean motions of the principal arguments, he obtained 
the corresponding part of the motion of the node by a method somewhat 
similar to that used by Dr. Hill for the perigee. One of the main difficulties 
in the latter case — the reduction of the equations to a suitable form— does not 
occur in the case of the node, and there is no indication in the single paper 
which Adams published on that subject in the Monthly Notices for 1877 that 
he contemplated the use of moving rectangular coordinates. But that he 
was investigating the properties of the latter may be inferred from another 
paper on the lunar theory in the same volume of the Monthly Notices. 

During the last six years I have been attempting to develop the ideas 
contained in the " Researches " by calculating the coefficients of terms with 
certain definite characteristics.* Dr. Hill had obtained those which had the 
characteristic unity, that is, which were functions of the mean motions of the 
Sun and Moon only, and also that part of the motion of the perigee which 
was a function of the same quantities ; Adams had done the same thing for 
the motion of the node. It therefore remained to obtain the general equa- 
tions, to put them into forms suitable for calculation, and to show how the 
other parts of the motions of the perigee and node might be obtained. 
Experiments were made with the inequalities whose characteristics are the 
first, second, and third powers of the ratio of the mean parallaxes of the 
Sun and Moon, and the same powers of the eccentricity of the Moon. The 
forms of the equations which were there used were, however, troublesome, 
chiefly from their liability to produce errors in the actual calculations. In a 
paper which will be hereafter referred to, for the sake of brevity, as the 
<! Investigations," f I showed how this difficulty might be avoided without, I 

* The " characteristic " of any part of a coefficient is defined to be that part in its 
expression which consists of powers and products of the eccentricities, the inclination, and the 
ratio of the mean parallaxes. The other factor is a function of the ratio of the mean motions, 
and it also depends on the nature of the coordinate used. See Chap. I. §15, below. 

t "Investigations in the Lunar Theory," Amer. Jour. Math. vol. xvii. pp. 318-358. 



Riff 

'.lit 



Motion of the Moon. 43 

think, causing any increase in the labour of making the calculations, and it 
has certainly diminished the actual time required for their performance. In 
fact, at least three fourths of the calculations might be performed by a com- 
puter whose stock-in-trade amounted to little more than a thorough know- 
ledge of logarithms. An effective control over the computations can be kept 
at almost every step ; and as the operations which would be turned over to 
the computer are always the same, he would soon be able to do his work 
with very little supervision. 

It is intended to apply the method so as to completely solve the problem 
of the Moon's motion as far as it is affected by the Sun and Earth alone, the 
action of the" Moon only on these bodies being included, and the three bodies 
being treated as particles of equal masses. The degree of accuracy aimed at 
is that the coefficients of all periodic terms in longitude, latitude, and parallax 
shall be included which are greater than o"'oi, and that they shall be correct 
to this amount. The number of terms required is undoubtedly very great. 
The calculation of coefficients up to the sixth order inclusive with respect to 
e, k — the lunar eccentricity and inclination — will be necessary ; those of the 
seventh order may be replaced by their elliptic values. The corresponding 
orders for ef, a — the solar eccentricity and the ratio of the mean parallaxes — 
are deduced from the fact that c /2 is roughly of the order e 3 or k 3 , and that a 
is of the order e 2 or k 2 . 

To obtain the coefficients with the above-mentioned degree of accuracy 
it will be necessary to calculate terms contained in about one hundred 
characteristics. These will include about five hundred periodic terms, and 
will require the actual calculation of perhaps two thousand separate co- 
efficients. The results now published contain the terms present in fourteen 
characteristics — that is to say, about one-seventh of the whole. Notwith- 
standing the fact that the number of terms contained in the higher charac- 
teristics is much greater than that in the lower ones, the work done so far 
probably amounts to more than one-fifth of the whole. This is due to the 
fact that a much higher degree of accuracy is required for the lower terms 
than is actually necessary to obtain the corresponding coefficients correctly to 
o" - oi ; the presence of small divisors causes a loss of accuracy, which has to 
be continually borne in mind in judging of the number of places of decimals 
which are to be calculated. For example, the term, the mean motion of 



44 



Mr. Eknkst W. Blown, Theory of the 



whose argument is twice trie difference between the mean motions of the 
perigee and node, requires that the calculations be actually carried three 
places of decimals further than would be necessary for a term of the same 
order with no small divisor. Fortunately, however, the majority of the terms 
which cause the most trouble, due to the presence of small divisors, are those 
which contain both ef, a in their characteristics, and therefore the number of 
characteristics of this nature to be considered is much smaller than would 
otherwise be the case. 

The theory will be an algebraical one throughout, with the single and 
important exception that the numerical value of the ratio of the mean motions 
of the Sun and Moon is substituted. The reasons for this may be briefly 
stated as follows : — First, slow convergence of the series which represent the 
coefficients arranged according to powers of m or m* e, k, e', a, takes place 
mainly along powers of m ; secondly, the value of m is known from observa- 
tion with great accuracy ; thirdly, estimates would have to be made of the 
values of the constants e, k used in this theory from the values of differently 
defined constants of eccentricity and latitude used in other theories ; fourthly, 
wery little, if any, extra trouble is caused by leaving e, k, e', a arbitrary. 
Thus the theory, while remaining to a large extent algebraical, will possess 
all the advantages of a purely numerical theory. It may be also mentioned 
that, by combining the results of this theory with Dr. Hill's modification of 
Delaunay, it can be effectively used for researches by the method of the 
variation of arbitrary constants. The procedure is. intrinsically contained in 
my paper " On the Theoretical Values of the Secular Acceleration in the 
Lunar Theory " in the Monthly Notices for March 1897. 

Of the four chapters which are now published, the first contains the 
whole theory, with certain exceptions, so far as it is necessary for the con- 
tinuous development of the numerical results. The exceptions are those 
parts of the theory which refer to numerical results previously obtained and 
which are not necessary for those which follow. For example, all details of 
purely theoretical interest are omitted, and no account is given of Dr. Hill's 
method for the determination of the intermediate orbit used here (which I 

• As usual m—n'ln, m=n'l(n-n'), where n, n' are the observed mean motions of the 
Moon and Sun. 



Motion of the Moon. 



45 



have called the "variation curve"), or of those methods used by him and 
Adams for finding the principal parts of the motions of the perigee and node. 
All that is necessary is a quotation of the numerical results, and they will be 
found in their proper places with the sources from which they have been 



obtained. 



The following is the table of contents :- 



Chapter I. — General Development of the Theory. 

Section (i). An investigation of the disturbing function used, with the 
necessary corrections. 

Section (ii). The two forms of the equations of motion. 

Section (iii). Development of the disturbing function according to powers 
of i /a, z, e'. 

Section (iv). The form of the solution. The general system of notation 
adopted to represent the coefficients, arguments, &c. 

Section (v). Method of solution. Preparation of the equations of 
motion. 

Section (vi). Exact definitions of the arbitrary constants used in the 
theory. 

Section (vii). Methods used for the solution of the equations of condi- 
tion satisfied by the coefficients. The long and short period terms which 
give rise to small divisors. Manner of obtaining the higher parts of the 
motions of the perigee and node. 

Section (viii). Details concerning the numerical calculations and the 
methods used to verify them. 

Section (ix). Transformation to polar coordinates. 

Chapter II. — Terms of zero order* Numerical results. 

Chapter III. — Numerical results for terms of the first order. 

Chapter IV. — Numerical results for terms of the second order. 

Future chapters will contain the terms of the third, fourth . . . orders. 

With regard to the calculations, no trouble has been spared to secure 
correctness. Errors are of two kinds, those which are merely numerical and 
those which are partly algebraical — i.e. due to the use of a wrong series of 
factors or to the omission of some series of terms. To test the former, 
equations of verification were computed at every step, and the nature of the 
method rendered these very numerous. An error of the latter kind may 

* The word " order" here and elsewhere refers only to e, e', k, a, and not to m. 



46 Mr. Ernest W. Brown, Theory of the 

escape such, a verification, and will generally produce a large discordance ; the 
results were therefore tested by a rough comparison with those of another 
theory — say, that of Delaunay. One searching test of this kind has been 
applied to the majority of the terms now published, namely, the compa- 
rison of the motions of the perigee and node deduced therefrom with those 
deduced by Hansen and Delaunay. 

The results obtained so far point to certain appreciable errors in the 
theories of Hansen and Delaunay. Dr. Hill has shown * that the last two 
terms of Delaunay's expression for the part of the motion of the perigee 
which depends on m only are wrong. The part of the motion of the perigee 
which depends on e 2 I have calculated in two quite different ways, and it 
appears that the last two, if not the last three, of these terms of Delaunay's 
expression are seriously erroneous.! An error in Delaunay's expression for 
the part of the motion of the node which depends on e' 3 was actually traced 
down to an error of transcription in his theory 4 Hansen's theoretical value 
for the annual motion of the node appears to be at least one if not two seconds 
in error.§ Professor !Newcomb in his discussion of the results obtained for the 
coefficient of parallactic inequality || considers that Hansen's value is about 
o"'30 in error : this amount, though small in itself, is of importance if the 
coefficient be used to obtain the solar parallax. I hope before long to finish 
the computation of this coefficient so as to obtain it within o //, 02, and, at a 
later period, the values of the annual motions of the perigee and node, so far 
as these depend on solar action only, correct to o"'Oi. 



CHAPTER I 

General Development of the Theory. 
Section (i). — The Problem of Three Bodies and the Disturbing Function. 

i. The disturbing function used by all investigators except Hansen 
gives only a portion of the inequalities produced by the Sun in the motion 
of the Moon. This form of the disturbing function will be used below, and 

* Annals of Mathematics, vol. ix. pp. 31-41. 

f Monthly Notices, vol. lvii. p. 336. % Ibid. p. 341. 

§ Ibid. p. 340. I Astron. Jour. vol. xv. p. 167. 



Motion of the Moon. 



47 



therefore we shall, in this section, investigate the small additions which must 
be made to it in order that the whole of the effect of the Sun on the Moon's 
motion may be obtained. The method of this section is similar to that of 
Prof. Xewcomb in " The Actions of the Planets on the Moon " (Amer. Eph. 
Papers, vol. v). 

Let X, Y, z, r be the coordinates and distance of the Moon referred to 
axes fixed in direction and passing through the centre of the Earth ; x', y' , 
z', i J , those of the Sun referred to parallel axes through the centre of mass of 
the Earth and Moon ; M, E, m' , the masses of the Moon, Earth, and Sun 
reckoned in astronomical units. The ^-coordinates of the Moon, Earth, and 
Sun referred to parallel axes through the centre of mass of the three bodies 
are respectively 



EX _ m' x' 
E+M m' + E+M' 



_ MX _ ml ri_ (E+M) x' 

E+M m' + E + M' m' + E+M' 



with similar expressions for the other coordinates. 

Let T be the kinetic energy of the system relative to the centre of mass. 
Then 



2 t=m\ ( J- - dJC - 



' \ \ E+M dt m' + E+ M dt 
M dX 



dx \ 2 



+ 



"\\ 



here 



7 

+e\( m dx +- ?*'_^V+ 

1 \E+M dt ^m'+E+M dt) + " - 
\\m' + E+M dt) ^ • • • I" • • • , 



+ 



, — EM 
Pl E+M' 



/j. 2 = 



'm' + E+M' 



Let F be the potential energy of the system. By Lagrange's equations 
we then have for the Moon's motion relative to the Earth— 



ft 



d*XJdF 
dt* dX' 



H-\ 



cPYjdF 

"if- ar 



hi 



d 2 z = oF 
dfi dx ' 



and for the Sun's motion relative to the centre of mass of the Earth and 
Moon — 



Hi 



d-x' dF 



H-2 



dh,' dF 



' dt- ex' ' r " z dt- 6y 
F being expressed in terms of X. Y. z. a/, y' , z 



8V dF 

at 1 o~ 



4 8 Mr. Ernest W. Brown, Tlieory of the 

Let r{ be the distance of the Sun from the Earth, A its distance from 

the Moon. Then 

„ EM , Em . Mm' , 

1< ■=. 1- — r H — — , 

r t{ A . . 

and 

r 2 =X 2 +F 2 + s 2 : r'^x'z + y't + z'*, 

where 

S is therefore the cosine of the angle subtended by A at the centre of mass 
of the Earth and Moon. 

2. We shall first consider the effect of the Moon's motion on the Sun 
and determine the deviation of the Sun's motion from an elliptic orbit in the 
plane of reference. For this purpose we may omit the first term of F, since 
it does not contain x' , y', z' . Then 

^ jE+M W a/ 
Expanding i/r/, i/A in powers of r//, we obtain 

F i '_L^j.w\r i x EM ri (3&-i\+ 

_-(m +E+M)^ J + lWTWyi r , 3 [ 2 $ 2 )+ • • • _• 

The order of the second term in comparison with the first is in the ratio 

Mr % : Er' 2 =i : 12,000,000 

approximately. The order of the next term is in the ratio Mr* : Er' ? >, a 
quantity which may and will be totally neglected. A sufficient correction 
to the elliptic motion of the Sun about the centre of mass of the Earth and 
Moon may therefore be obtained by using 



(m+E+M) w+MYir .ja- 2 ) 



the disturbing function, and substituting for the coordinates of the Mcon 
their elliptic values, together with the principal inequalities due to the Srn, 



as 



Motion of the Moon. 49 

as found in Chaps. II., III. below. If ri be the observed mean motion of the 
Sun, we define the mean distance a' by the equation 

3. Next, for the motion of the Moon we have, on expanding F and 

rejecting the useless term m'(E+M)//, 



/a, r 



^ EM \r/ A 



_E+M , m'r 2 
r 



^^->e1m^-Is) 

&-EM+M2 tY3S S 4_ I S S2 + 3A 
(E+Mf r' 2 U 4 8> 



+ (E+Mf r'\ 8 b 4 " + ¥ b J + • • • J * 

The expansion is carried as far as will be necessary, since we shall 
neglect quantities of the orders r 4 // 4 , M> s /E? /3 . 

The force-function which we shall use is that ordinarily used, namely, 

r LvV 2 -2rr'S+r 3 r' r' 2 J 

r r i [_\2 2) v\2 2 J 

<(^-¥^)+--] - <■> 

In this, the elliptic values of x', y 1 (Y=o) are substituted and m' is put 
equal to n n a' s instead of n'' 2 a' 3 -E-M. It is necessary to consider what 
corrections must be made when Si is used instead of F/ /J , 1 . 

4. The corrections are of three kinds and are so small that we need only 
consider their effects to the first order of the disturbance. 

(a) Correction due to putting n' 2 a' 3 =m' instead of m' + E+M. We 
must add to 8 the terms 

r' 6 \_2 2 r \2 2 /J 
liOYAL AsTIiON. SOC, VOL. LIII. T 



5 o Mr. Ernest W. Brown, Theory of the 

This correction will be sufficiently accounted for if we multiply all the 
inequalities due to the Sun by 

._ E+M __ _ i 



330 000 

approximately.* 

(b) Correction due to using the elliptic instead of the true values of the 
solar coordinates. We must add to 8 the term 

»'v{^0s.-I)], 

where S operates on a/, y', z', and 8/, St/, In! are the perturbations of a-', y', z', 

as found in § 2. 

(c) Correction due to the difference between F/^ and Si. Since the 

2 ., 3 

terms in F/^ or S3 involving n'V 8 -^ 1 ^ j give inequalities having the factor 

U/a') J , where a is the " constant " of the Moon's distance, this correction may 
be partly made by multiplying the inequalities having the factor (a/a')' D y 

(E-MV 
\E+MJ ' 

To the order considered here, there will then remain to be added to 
S3 the term 

^ ri rn EM »V35 S 4_f5 s4 + 3Y 



n a —^, 



5. "The method of procedure is therefore as follows : — The values of X, 
Y, z are first obtained by using q with m'=n' 2 a' 3 , z' = o and elliptic values 
for x', y'. With the values of X, Y, z thus formed we compute lx' , h/ , W 
by means of the disturbing function in § 2. The corrections to, X, Y, z, 
noted in (a), (b), (c) above, are then easily obtained. The first step is that 

* Except for the mean motion of the perigee where the second and succeeding terms, 
which are of the order of the square and higher powers of the disturbing forces, are a little 
Greater in actual value than the first term. The correction may be made with sufficient 
accuracy by multiplying the mean motion of the perigee by 



2 to' 2 ' m' 2 m' 



See Monthly Notices, 1897 June. 



Motion of the Moon. 5 l 

which very frequently bears the name of the " Lunar Theory." It will be 
noticed from this investigation that the elliptic values of the solar coordinates 
to be used are those referred to the centre of mass of the Earth and Moon. 

Section (ii). — 77*6 Equations of Motion. 
6. Let 

x, y, ~, be the coordinates of the Moon, referred to rectangular axes through the Earth's 
centre, of which those of x, y are in the plane of the Sun's orbit (supposed fixed 
and elliptic), the positive direction of the moving .r-axis being constantly 
directed to the mean place of the Sun ; 
r i =x i + y i + z i ; 
p->-=x? + y* ; 
n, n! , the observed mean motions of the Moon and Sun ; 
r', e', a', the radius vector, eccentricity, and semi-axis major of the Sun's orbit (§ 5) ; 
v, the solar equation of the centre ; 
Si=aj cos v + y sin v. 

So that 

■rr'8=Xx' + Yy' + zz' =r'(x cos v + y sin v)=r'S 1 , 

and 



.■2— n i 



=p»+.' 



Also, representing V—i by 1, let 



u=x + yi, s=x — yi, ns=p- ; 
ri E + M 



m= 



n — n'' (n — n')- 

£=exp. (n — n')(t — t )i, 

n 1 d yd 

(n — n')i dt d'C, 



where t is a constant to be defined later. 

The equations of motion, referred to the moving axes and with the force- 
function a (§ 3), are 



d'x ,dy , 2 'dp, 

dt 1 dt ex 

d 2 y , ,dy „ op 

df- dt J ay 

dh _o P 

dt* ?-~ ' 



52 



Mr. Ernest W. Brown, Theory of the 



Let 



2 2 



and transform to the independent variables u, s, z and the dependent variable 
t The equations become 



D 2 w+2m2)«=- 
DH = - 



2 do/ 

(n — n') 2 ds 

2_ 9o_' 

(re — «') 2 dn ' 

_i 8 a'. 

(n—n') 2 3s 



where, by Sect, (i), 



^-{us + z'r L(r'*- 2 r'S 1+ M 8 + ^ r> r"J T 2 



This gives, on expansion according to powers of i//, after some trans- 
formations, 



(re — re') 2 (us + z-y 4 



W 



here 



Oi==3 m " 



+ 



S S 1 3- 3 S 1 («« + ^)" 



m 2 a' 4r 



+ 



a r 

m 2 a^_ 
a 12 ' r' s L 4 

m 2 a' 6 



(2) 



a' 8 ' r' 6 |_ 4 



' 6 _3g 1 5_3S Sia ( MS + s 2 ) + iS Si ( Mg + ^ 



+ 



suppose, where «, is the part of s? x which involves u, s, z to the degree 
</ ; Si is of the first degree with respect to u, s. It will be noticed that w 2 is 
zero when e?=o. 



<p 



Motion of the Moon. 



The equations may now be written— 

(D + m) 2 u + xn 2 u+^m 2 s — 



(us+z 2 )i ds 

ks 3 q , 



(D— m) 2 s + m 2 s -f 3 m ? u — — v — 

2 2 (ms + s 2 )'* clu 



(Z>»_m*) 5 - ,— -f = -A?> 

(mS + S 2 )* J OS 



53 
(3) 

(3') 
(4) 



These are the fundamental equations in the theory. Since u, .<? are 
conjugate complexes, either of the first two equations is sufficient ; we shall 
use the first in the calculations. 

7. Homogeneous Form of the Equations. — Multiply the three equations by 
Da, Du, 2l)z respectively, and add. We obtain 

D\Du . Ds + (Dz) 2 + 3 m 2 (u + s) 2 -m ,i z 2 + — 1 

L 4 (us+z 2 yj 

L ds du dz J 

Since S3 1 is a function of u, s, z, r', v only, and since ?•', v are supposed 
to be known functions of the time, 



du ds dz dt 



and therefore the right-hand member of the previous equation 

= d ^-Dt-D &> = .-- ' d ^-D Si} 
dt (n—n')i at 

=D' Q^-D&^DiD-WQi)- Si ,]. 

where D~ l denotes the operation inverse to D [i.e. integration with respect 
to £ followed by a division by £), and X^'S^ denotes the operation D performed 
on Si j only so far as £ occurs in r', v. 

With this form of the right-hand member we can integrate and obtain 



Du . Ds+(Z)z) 2 + - i m 2 (u + sy-m 2 z 2 + 



{us + z'f 



--C'-Si,+D-\D'i 



(5) 



Now add this to the sum of (3), (3'), (4) multiplied by .<?, it, 2; respec- 
tively. Since, by Euxer's theorem, 



'IS (« (Z .._.. ' 



54 Mr. Ernest W. Brown, Theory of the 

we obtain an equation which may be written — 

-- D%<us + z*)-Du . Ds-(Dzf-2m(uDs-sDu) + ^m%u + sf-3rn 1 z a 

Also multiply (3) by s, (3') by u and subtract ; multiply (3) by z and 
(4) by u and subtract. The two resulting equations are 

D{uDs-sDu-2mus)+tm\i?-s i )=s d ^-u d ^ ( 7 ) 



2 



B(uDz-zDu)-2mzDu-m !> uz-lm i z(u+s)=z ®-k-± u 4jj± - ( 8 ) 

Instead of the last we may obtain the more symmetrical form, 



D[(u - s)Dz - zD(u — s)] — 2mzD(u + s) - m 2 «(w - s) 

\ os ou J 2 oz 



(8') 



The equations (6), (7), (8) will be called the homogeneous equations. 
Their left-hand members are homogeneous and of the second degree with 
respect to u, s, z, while their right-hand members are, abstraction being made 
of C, of the 2nd, 3rd . . . degrees with respect to the same variables.* 

Section (iii.). — Development of S a . 

8. In the last section, the development of S?j according to powers oil /a' 
has been given ; the result is there numbered (2). We must now further 
develop it according to powers of e' and z, which are small quantities of the 
first order. The development will not be carried beyond quantities of the 
orders — 

a 3 a 2 , a 2 „ 2 a , 3 ,., 

„'3' „<a e ' "di" ' a' ' ' 

G, " Or & Lb 

By the definition of Sj we have 

S { =x cos v + y sin v 

=~~ (ue~" +se v ') 
2 

* Fuller explanations of the transformations in the section are given in the Treatise cm 
the Lunar Theory, Chap. II. (iii.). 



Motion of the Moon. 



55 



Substituting in (2) and remembering that 

Si 1 =<^2 + °>3 + w 4 + <°S> 

we have, to the orders of small quantities just mentioned, 



3 
.4 



(w 2 a 2 4-s 2 a 2 ) + -t«B 2 — « 2 b 2 L 



w 3=- 7 



u 



.3/,/ 2 



(w 3 a 3 + s 3 a 3 ) + j?(w 8 sc 3 + ush 3 )—^uz% - J ss "c 



«L 2 f35 
4 a' 2 L64 



"4—^1 >-■ (** 4 fl4 + * 4 a 4 ) + 5 , (m 3 «c 4 + ms 3 c 4 ) + - J w 2 s 2 f 4 
~ 2 ,64 16 3 2 



S 2 f -„ S U% + * S « 2 C 4 + 9 M6'[' 4 j 



where 



- 6 -o (W 5 + « 5 ) + - 3S „ (« 4 « + MS 4 ) 4- J 5 /«¥ + W*S 3 ) 

,128 128 64 



2 .,. 3 



(Ji=- ■--. « '"" 



r " 



t 1 2= ,- 3 -r> tl 4=;,,- ) 






a 

c -i= .-, ( 
r 



and a 2 , f) 3 , . . are the values of n 2 , h 2 , . . when — 1 has been put for 1. 

9. The quantities a 2 , i 2 , are to be expanded in powers of e' ; they 

are well-known elliptic expansions and they have been computed by several 
investigators. They may be conveniently obtained from the expressions 
given by Delaunay in chap. ii. vol. i. of his Theorie de la Lune by giving to 
the angle, there called a, suitable values, or from the tables of Cayley {Mem. 
R.A.S. vol. xxix., Coll. Works, vol. in). 

Putting Exp. l'i=£ m (see § n, below), 

where I' is the solar mean anomaly, we obtain 



n 2 = 



= _5 e /a + i3 e /4 
2 16 

+ . £ 2m 

+ _ r _ e '4£4m 

24 

1280 



384 



V2 10 128 J 

I 7g'2_ II 5 e '4V^2m 
2 " 6 J 



+ l 4 8 768 J C 



533 
16 



e' 4 C" 



228347^ 
3840 



56 Mr. Ernest W. Brown, Theory of the 

2 2 8 \2 16 128 ;\. J 

4. .e' 3 C* m 4. l6 V 3 £~ 3m , 

4 

4 lr 6 '2^2m + 53 e '2^-2m 



C :< =i 



+ 2 3g'3^3m 4. 7 7 e '3^-3m j 

12 6 



a 4 =i- 3 e'£ m + I 3 e'£- ra 

2 2 



The values of H„ c„ . . . are obtained from those of a. 2 , c„ . . . by putting 

i/£for£. 

Section (iv). — Form of the Solution. Notation. 

10. Let Fbe the true longitude in the plane of X Y or x y reckoned 
from the fixed axis of X and 4, the latitude above this plane. ^ Also let 1>, 
Z, f , F denote the same angles as in Delaunay's theory, Chap. xi. vol. ii. i.e. 
let 

D=(«-w')« + c-e'=Half arg. of the "Variation," 
I =cnt + e-m =Arg. of the Principal Elliptic Term, 

Z' =«'< + £' -V = „ „ " Annual Equation," 
1?—g n l + t -.$ = „ „ Principal term in Latitude. 



Motion of the Moon. 



57 



Here, as usual, e, sr, 6 are the values of the mean longitudes of the M 
and of its perigee and node at time t=o ■ e' is the mean longitude of the 
at time t=o and to' the (constant) longitude of its perigee; (r- 
( i -g)n are the mean motions of the perigee and node. 

We then have 

X= P cos V, Y=p sin V, z=p tan $=r sin <j>. 

Whence, as we shall put £ = exp. Di, (§ n), 

x—pcos(V-n't — e')=pcos(V-nt — e + D), 
2/=psin(F-n7— e')=p sin (T-««-e + D) ; 



oon 



:un 



w =pexp. (V—nt — £ + D)i 
utr l =p exp. (F— m« — c) (, 



s =pexp. -(F-w<-e + D) t 
s£=pexp. — (V— nt — e) t 



(9) 



It is well known that, with the limitations here imposed, r, V- n?-e, <j> are 
expressible by sums of periodic terms whose arguments are (algebraic)' sums 
of multiples of the four angles 1), I. /', F. Hence ,i-, y, z are Expressible in 
the forms 



X\ COS") 

y[=&%A i:F:Clir sin' (iB+pl + rl 1 + qF), 



h P, q, r=o, 



where a is a linear constant and A a coefficient ; the sign of summation 2 
denotes that the sum of all such terms must be taken. ° Remembering the 
definition of u, *, it is evident that u, s, a may be therefore expressed in 



the form 



za ^ A i,p,<!,r exp. (iD+2)l + rl' + qF) 



with certain limitations which will be set down later. 

There are four sets of notations required. First, for the exponentials : 
secondly, for the constants of distance, eccentricity, latitude and ratio of the 
mean parallaxes ; thirdly, for the numerical parts of the coefficients (the>o 
parts are functions of m only, and the numerical value of m is used thrcno-h- 
out) : fourthly, for the terms of u, .*, z which are of a particular order. 

Royal Astrox. Soc., Vol. LTif. ,. 



5 8 Mr. Ernest W. Brown, Theory of the 

ii. Notation for the Exponentials.— Recalling that m= ft' / (n~-n'),^emay 

evidently write 

B = (n-n') (t~t ), 

l=c (n—n 1 ) (t—t x ), 

l'=m (n-n')(t-t 3 ), 

F=g(n-n')(t-t i ), 

where 

c (w— n')=cn, g (n—n')=gn, 

and the signification of t Q , t 1} t 2 , t 3 is obvious. 

We have, in Section (ii), defined K by the equation 

£=exp. (n—n') (t — t e ) i, 

and we now give to t in this expression the same meaning as it has in D, 
so that £=exp. Di. 

Let, for a moment, 

£ c =exp. c{n-n')(t-t{)i.. 

Remembering the definition of the operator D we have 

where i, j, p are positive or negative integers. 

Now. in the method pursued here, we shall always proceed by equating 
to zero the coefficients of like powers of K, K° c in equations which consist of 
such expressions as that just written down, and it will never be necessary to 
substitute the value of c in the indices ; its value is only substituted in 
the coefficients. The above equation shows that the coefficients will be the 
same whether we write K%\ or, f s \ Further, as the suffix c always occurs 
in the index whenever it is present as a suffix, the suffix is unnecessary for 
purposes of distinction and we shall omit it in future. The same remarks 
apply to 

/ m =exp.»A £' ig =exp.5Fi. 

AVe may therefore put 

exp. (tD + P l + rl' + q-Fy^"*™***, 

the index of K always denoting the coefficient of t in the corresponding 



argument divided by n~ri. 



p 



p. 



Motion of the Moon. 



59 



12. Notation for the Arbitrary Constants and the Parameters. — There are 
six arbitrary constants present in the solution. Three of these— e, ct, 0, or 
f o, t ly 4 — have already been considered ; they are contained in D, I, F, or in 
£, £ c , K g - The other arbitraries to be used will be denoted by 

n, a, e, k, 

connected by one relation. The exact definitions of c, k,— the constants of 
eccentricity and inclination— will be found in Section (vi) ; n has been defined 
as the observed mean motion ; the linear constant, a, is connected with n, e, k 
U+Mbj a relation which will be defined in the same Section. In elliptic 
motion, this relation would be n 2 a:'=E+M; in the actual case the relation 
differs a little from this. 

The parameters in powers of which expansion will be made are 

m, e, e', k, a= a 

a' 

The numerical value of m is used, but the other parameters are left arbitrarv. 
13. Notation for the Numerical Coefficients. —From what precedes, it is 
evident that u, s, zi may be expressed in the form 



=a2.4. 



,rV + * 



or, as it will be more convenient to write it, 



>P, q, r= + co 



The latter form has the following properties, which are easily deduced from 
the known properties of the expressions for V, r, <£. 

(a) sK is deduced from nXr x by putting i/f for % ; 

if i/Cbeput for ? in the expression for zt, the coefficient merely 
changes sign ; 

(b) i is odd or even according as A contains odd or even powers of a ; 
(0) g is even in the expressions for u%-\ hK and odd in that for zi ; 

(d) A is of the order e w e' 1 " k 1 *' at least, and it contains higher powers 

of e, </, k which differ from \p\, \r\ , \q\ by even integers. 
14. I now give the general notation adopted throughout. It is devised 
so as to represent every part of every coefficient. 



65 Mr. Ernest W. Brown, Tlieory of the 

The general term in uf 1 or z t will be expressed by 

a(€ p+ »'E' p V +r V r '^ +s '/fc'' J 'a*') i e J,+2p 'e' r+2 '''k s+2? 'a , 'C 2i;fc|,cira±M ( I0 ) 

where * 

p, q, r, /, q', r', s'=o, i, 2, ... , 

2i=o, ±i, ±2, ... 

The coefficient ( ) t above written corresponds to the upper signs in the index 

of ?• 

With the lower sign ofpc, interchange the indices of e, e' ; 

With the lower sign of <rm, interchange the indices of v, V ; 

With the lower sign of qg, interchange the indices of l,li. 

The sum of all such terms for all values of p, q, r, p', q', /, «', 2% gives the 
complete expression of uK" x or of zi. 

From the properties (a), (b), (c), (d) just given, it is evident that when 
q is even, (io) gives the general term of uK" x ; that when q is odd (io) gives 
the general term of zt, and that then 

that *r is deduced from uK~ x by putting i/Kfor ?; that J, zi are odd or even 

together. 

When 2i is odd, we shall frequently denote this fact by putting 22 = 2^, 
so that h will denote an odd multiple of ±-|. When an index of any symbol 
inside' (), is zero, the symbol is simply omitted. 

In the cases of the coefficients of the first order, namely, (e)„ (e% (?)<> 
(v')i, (*)<> ( £ 0» tne brackets ( ) will be omitted for the sake of brevity, as 
they are unnecessary. 

Particular Case.— In the case where p, q, . . . s' are all zero no letter 
would occur inside ( ),. This being inconvenient we shall denote the corre- 
sponding coefficient by a,. Thus, the terms independent of e, d k, a in uK~ x are 

denoted, by 

nSfiff 1 , i=o, ±i, ±2, . . . 

These are the terms of order zero (§ 15 below). There are no such terms in z. 

* No confusion will be caused by this new use of the letters r, r', since they only occur, 
in this sense, in the indices of rj, t/, «', and have positive integral values. 



Motion of the Moon. 
15. Characteristic and Order. — The factor 



6x 



of the general coefficient will be called the characteristic part of the coefficient 
or, briefly, the characteristic. 

The Order is the sum of the indices of e, e 1 , k, a. The order of the 
general term is thei-efore 

p f 2p' + r + 2r' + q + 2q' + s'. 

The word " order," as used here, is thus independent of m — a necessary 
restriction, since the numerical value of m is substituted at the outset. 

A few remarks and an example may make the notation laid down in § 14 
clearer. It will be observed that 

e, e' are always associated with e ; 
rj, ■>]' are always associated with e! ; 
k, k' are always associated with k ; 
a is always associated with a=a/»' ; 

while, as is well known, e is associated with the index c ; e' with the index 
m ; k with the index g ; and an odd power of a with an odd value of 21. 

If the numerical value of m had not been substituted at the outset, we 
could further have denoted the particular power of m involved bv inserting 
m J inside ( ),• ; the coefficient ( \ would then have been a positive power 
series in m, with a numerical factor for each power of m, which factor is always 
the ratio of two integers. 

The actual arrangement of e p+p ', e' 1 '', . . . inside ( ), is immaterial, but we 
shall, in general, retain the arrangement of § 14. 



Example. — The numerical part of the coefficient of £ 3+2m - 2 « 

is denoted by 

0/y&' 2 a)j. 



which 1 



ias 



the characteristic e'Va in ut~ l 



For here. p = o, p'=o : 



interchanged 






:2, /=i ; q=2, q'=o with the indices of L I: 
The series of terms in v.t~ x which will be 



found along with this term, are those of the same characteristic which are 



■I 



62 Mr. Ernest W. Brown, Theory of the 

obtained by putting f 1 for K and whose indices differ from the given index 
of K by even whole numbers. See § 27 below. They are 

ae'% 2 aS ii [(^V^ 2 4 l f 2f!+2m_2g +(W 3 ^«) i ,r i - 2m+2g ], 

where 

2 i i =±i, ±3, ±5, . . . 

16. Notation for terms with a given Characteristic or of a given Order. — 
It will frequently be convenient to specify such terms in a brief manner ; 
this may be done by means of a suffix attached to u, s, z. Thus all terms 
in u with characteristic e 2 may be denoted by u e >, those with characteristic 
(fa by u e , a , and generally, those with characteristic ^ by u K , 

The terms of a given order are denoted by numerical suffixes. Thus. u 3 
denotes all terms of the third order ; u , those of zero order ; and so on. 

Section (v). Method of Solution. Preparation of the Equations. 

17. The general feature of the method consists in the successive 
determination of the terms of orders o, 1, 2, . . . with respect to powers and 
products of e, ef, k, a in the coordinates u, z. As will be seen from the results 
of Section (ii), there are two methods of procedure — one by the use of the 
equations (3), (4), and the other by the use of the homogeneous equations 
(6), (7), (8). At the same time it may be stated that we need by no means 
confine ourselves to either of the two sets of equations, but use one or the 
other or both as may seem most convenient for the particular class of 
inequalities under consideration at any stage of the approximations. 

18. Terms of Order Zero. — These terms, the coefficients of which are 
functions of m only, constitute a closed orbit with reference to the moving 
axes which is really the primary or " intermediate " orbit in the same sense 
as the elliptic orbit of the older theories ; it may be called the Variation 
Curve since its principal periodic term is known as the " Variation." According 
to the notation of the last Section it is given by 

u^=B,%aff\ So^aSflL.iC* (") 

These values u , s Q of u, s constitute a particular solution of the 



equation 



(#4-m)% + -V 2 M + lm 2 s- K v=o (12) 

X ' 2 2 ft 6 



Motion of the Moon. 



63 



or, in the homogeneous form, of the equations, 

D\u8) — Du . Bs — 2m(uDs—sDu) + 1m\u + sf= C" 
D(uDs— sl>u — 2 raws) + 2-m\ii? — s' 2 ) = o 



(12') 



since here, Si 1 =o, 2=0. The constant C is a function of a, m, while a is a 
function of E+M, n, m. 

The values of u, s being substituted in the equations of either form (by 
preference, the latter), we obtain a number of equations of condition which 
suffice to determine aa t . In consequence of the presence of a, one of the a t is 
arbitrary; we put a =i. The whole theory of these terms and the numerical 
results have been fully worked out by Dr. Hill in his paper " Researches in 
the Lunar Theory" (Amer. Jour. Math. vol. i.) ; the results will be given in 
Chap. II. 

19. Terms of the First Order. — We put 



where 



u=u + u 1 , z=z u 

and neglect powers and products u u s t , z u above the first. In the general 
equations of Section (ii) we pxit 

53j=o, s=o, whenw 1 =M e ; 

Q, 1 =b> 2 , s=o, when u 1 =u l: , ; 

Q l =(j> s , 3=0, when u 1 =u^ ; 

g >l =o,u l =o, 'whenz l =z k . 



Further, when Si x =o). 2 we neglect e' 2 and higher powers, and when Sl x =a> 3 we 
put </ = o; in both cases u , s may be put for u, s in Q v 

The right-hand members of the equations thus consist entirely of known 
terms. Putting u=u li +u l . s=s + s u z=z L , and expanding the last terms of 
the left-hand members, it is easily seen that the equations (3), (4) for u u z t 
become 

D^ l -2Mz l =o ) ' 

where, putting p 2 = ? 'o'o, 



(13) 



2 



2Po J 



J7=%F,t 2! = 3 -m 2 +• 3 KM o'%;i I 
2 2 f , a ' 1 



( l 4) 



■■■n 



64 Mr. Ernest W. Brown, Theory of the 

and A consists of a ^-series with known coefficients, and \=e' or a. When 
Ul =u e , we have A = o. The sets of inequalities corresponding to u e , it e -, u a , z k 
can evidently be separately determined. In each case the appropriate expres- 
sions for u h s x , z u are substituted, and equations of condition for the unknown 
coefficients are obtained by equating the coefficients of the various powers 
of £ to zero ; they are then solved by continued approximation. 

In the equation for u e , the questions of the arbitrary constant of eccen- 
tricity and of the motion of the perigee arise, and in that for z k , the arbitrary 
constant of latitude and the motion of the node. ; these will be dealt with in 
their proper place (§§ 25, 26). 

The homogeneous forms of the equations may be considered in a similar 
manner. I do not give the developments here as they may be easily inferred 
from the more general treatment in § 22 of this Section. Further infor- 
mation is given in chapter xi. of the " Treatise on the Lunar Theory " and in 
the papers on the Parallactic and Elliptic Inequalities (Amer. Jour. Math, 
vols, xiv., xv.). 

The numerical results for the terms of the first order are contained in 
Chapter III below. 

20. Terms of the Second and Higher Orders. Development of the Equation* 
( 3 ) ? ( 4 )_ The terms of the first order having been obtained, we proceed to 
show generally how the terms of any order and characteristic may be found 
when those of lower orders have been calculated. We shall deal with both 
sets of equations as either may be useful in certain cases. In this section the 
equations (3), (4) are considered. 

They may be written, 

™ + m)«u+-Vur 4 3 m*»£ . f 1 - -""f-^-S" 1 -^ • • (15) 

v 2 2 f l OS 

m^ m ^- K "-- ld ^-i (16) 



2 OZ 



Suppose that it be required to determine all terms having the cha- 
racteristic X, say u M in u, and those of characteristic* X, say z k , in z. 

* Of course X can never be the same in the s-equation as it is in the w-equatioii. Conse- 
quently, as terms of different characteristics are never found in the same set of equations of 
condition, the equations (15), (16), are never considered together in finding any particular set of 
coefficients. 



Motion of the Moon. 6 c 

We put 

u=u (l + S l i^ + u„ 2=%, in (15), 

u=u + S,u^ z=%z^ + z x , in (16). 

Here 2u„ S\ contain all terms in u, z, except u , u kt z K , which will contribute 
to give terms with characteristic \. These expressions are substituted in 
the equations, which are then expanded according to powers of u-ti 0) z, all 
powers of u K , z k above the first and products of n k , z k with u M z^ being, of 
course, omitted. It will be remembered that r 2 = m + z\ 

Choosing out the terms which may produce terms with the characteristic 
X we find 

=Part, char". X, in \ -t l {D* + 2mD)3 < u -^if 1 
L os 

Po 3 (8V. u J S\ s J 4 u e Q 2\ Po J 

-5 ^y_3s^sA :! _ 9 pOr%_ IS /s^y ^ 

i6Vm / i6V*oi i6\w / s i6\s ) u 

4\ Po J «o 4 \ Po / «o 

+ (] - -. - (17) 

=Part, ehar c . A, in \ -tr\D°-+2mD)%u -^pJ £-1 

4 4 J 

+ ' ' J - - <i7') 

Z> 2 s J -2i¥s 1 =Part, char". A, in l-D^z- 1 8 J^ 

L * 2 cz 

+ _!Lr_3 2 ?M ^«; + ?M 

Po' L 2 p v w « y 
+ %« i5^y + i 5 /5«,y 9 5^ %l_3/s^y 

Po I 8 V «o / 8 ^ s () y 4 Mi) Sq J 2 V po / 

Royal Astron. Soc, Vol. LITE. 



]] ... ... (I S) 



66 



Mr. Ekxest W. Brown, Theory of the 



=Part, char . \, in 



-D^s, 



108, 

2 03 






... (i 



where 



P=%P£ %i 
B=%Bff' 



a KM °^ ' 






Q = 2,Q^=a 



Po' 



3f-3 






T=^ l TX ii ^^ KU °"' Ci 



(><> 



P=XP_ff ! , 



Po" 
R=%fi^' 



(i9) 



etc. 



the values of R, R being obtained from those of P, R by interchanging u,i \ 
6'oC, that is, by putting i/£ for £.* 

21. Some remarks on these equations are necessary. In the first place, 
the left-hand members of (3), (4) being linear with respect to u, s, and to 2 
respectively (exception being made of the terms containing k ; these will be 
considered immediately), the parts Uo + Xiip of u and % of z cannot contri- 
bute anything to terms with characteristic X. as far as the coefficients of these 
periodic terms are concerned. But the operators B 2 , D cause the coefficients 
.in u tl , z^ to be respectively multiplied by factors of the forms 

(2i+pc\-rmiqgf, (2i+pc + rm + qg), 

and c, g contain powers and products of e 2 , e' 2 , F, a 2 . Hence it will be 
necessary in some of the terms, whose orders are higher than the second, to 
include B\, Du„ B 2 z (J , in the equations. 

In o-eneral, all the unknowns are contained in the left-hand members of 
(17) or (17') and (18) or (18'), while the terms of the right-hand members 
are entirely known. Exception to the last statement only holds when we 
are determining an unknown part of c or g. It must be remembered that, in 

* In the Investigations, p. 327, where those expressions are used, one or two errors must 
bo noted. The factor a (there called a ) is omitted from the values of P, P, Q and the factor 
a* from those of R, R, T, S. The notations for B, R should be interchanged, that is R_ t put 
for /)';, to bring them into uniformity with those for P, P. 



Motion of the Moon. 



67 



reality, c and g are supposed to represent the complete values at the outset, 
but that, in forming the equations of condition, all terms of higher orders in 
the values of c, g, than those actually under consideration are neglected. 
Hence an unknown part of c or g will, in certain cases, arise from the terms 
containing the operator I) in the right-hand members. These cases are more 
fully considered in the next Section. 

As to the terms involving SZ h since Q t is of the first order at least, apart 
from the order of the terms in u, we can evidently substitute u, + Xu lx for v 
and %z^ for z therein. 

The rest of the terms arise from the expansions of ^I*, ^ in powers 

of %, %• Those containing the first powers of these quantities are 
omitted, since they evidently cannot produce terms with the characteristic X. 
It will be noticed that the terms are written in two ways, (17) ( 18) and 
(17'), (18'). } 

If we take the first forms, namely (17), (18), for calculation, the pro- 
cess is to calculate each Xuju a , XzJp Q (which consists in a single easy 
multiplication of each u^ 1 by . % £/ p * and of each z >L by i/p Q ), to form the 
various products inside the parentheses, and finally to multiply the resulting 
series by Kii^/p* and K /p* in the respective equations. For this purpose 
the values of 






a 
Po' 



ku £ 



a/V 



expressed as even-power series in £ are given at the end of Chapter II. 

If we take the second forms, namely, (if), (18'), the various multipli- 
cations which have to be made are evident, and the values of P, Q, . . 
necessary for this, expressed as power series in £, are given in the same 
place. There can be little doubt that the first forms give shorter calculations, 
and they have a further advantage in the fact that the only trouble necessary 
for transferring to polar coordinates consists in the calculations of powers of 
XuJuq, Xzjpo (see Section ix), and this labour will therefore have already 
been finished. Indeed, the second forms would not have been given here at 
all, were it not for the fact that I failed to see the great advantage of the first 
forms for the higher inequalities, and consequently used the second forms in 



68 Mr. Ernest W. Brown, Theory of the 

the calculation of all the second order inequalities and of a few of the third 

order.* 

A great advantage of these equations is that the chief labour— the multi- 
plication of series— can be easily arranged for an ordinary computer, and 
much of the merely mechanical labour may thereby be distributed (see 

Sect. viii). 

The numerical results for terms of the second order are given in 
Chapter IV. below ; those for terms of the third and higher orders will be 
given in chapters to be published hereafter. 

22. First and Higher Order Terms. Homogeneous Equations.— The 
separation of the homogeneous equations into known and unknown parts is 
effected in a similar manner. The substitutions for u, z are the same as in 
§ 20. If the homogeneous equations be used for actually finding the co- 
efficients, the forms obtained in equations (20), (21), (22) are of no assistance, 
except as a guide ; this will be evident by a glance at §§ 32-36. They have 
been, however, almost exclusively used for verifying the results obtained 
from the equations of § 20, and they are given mainly for that purpose here. 

They are 

2)2 ( M(A + 8oU j _ Bu Ds x - Ds Du x - 2 m(u Ds x - s Du x + u x Ds - s x Du ) 

+ lm 2 (u + s )(u x + s x ) 

2 

=Part, char=. A, in ["<?'- S( ? + iH + ^S 1) 

L 3= 2 

-(D%) 2 - 3 m 2 (%) 2 }l ... ... ... (20) 

2 J j 

D(u Ds x -s Du x + u x Ds - s x Du - 2mu s x — 2ms Q u x ) + T,m%u Q u x —s s x ) 

=Part, char*. X, in [&&±-u d ®-± 

' |_ as ou 



(*0 



U V, 



* From the remarks just made, it might have seemed an improvement to put u 
s=s v, z= Po w and to find v, v, w only. This, however, only throws some of the labour of 
forming u» /w , etc., on to the solution of the equations of condition ; the latter process is far 
less mechanical than the former and much more liable to error, and there will be no saving 
of labour. 



Motion of the Moon. 



69 



D(u JDz x -s A Du ) - 2mz x Dic - voSi a z x —^mh k (u + s ) 

2 



2 oz 



=Part, char=. A, in fs °&-l-J 
L c« s 

- {Dtfu, . Dtz^-1%^ . D% Ul )— 2m% . DSt^-m 2 ^. % 



(22) 



As before, the left-hand members contain the unknown terms. The 
right-hand members consist entirely of known terms, except when new parts 
of the motions of perigee or node are under consideration. In certain cases 
it will be necessary to suppose u , s to be included in %u M Xs^ ; see the 
remarks at the beginning of § 21. In the terms involving £1 ; we may 
substitute u^ + Xu^ Zz^ for u, z, respectively ; the method by which the 
calculations are to be actually performed will be given in Section (vii). 

In the case of the first order terms, Zu^, Xz^ are both zero, and the 
limitations of a t are the same as those given in § 19. 



Section (vi). Definitions of the Arbitrary Constants. 

23. Of the six arbitrary constants of the solution, three have already 
been defined, namely, the three angular constants in the arguments D, I, F. 
A fourth, n, has also been defined as the observed mean motion. It remains 
to give an exact definition to a, the linear constant (which replaces the mass 
E+M), and to e, k, the constants of eccentricity and inclination. 

24. Definition of a. — In elliptic motion, a is defined by means of the 
relation n 2 a 3 =E+M. For many purposes this is much the most convenient 
definition even when we proceed to determine the solar inequalities in the 
lunar theory. But in the theory developed here, the calculations may be 
materially shortened by a different definition of a. 

The value of u is given by 



i( £ l =aSjffl(£ 2i j 



1 = 0, 



From the form of this it is evident that either a or a may be chosen to be 
anything we wish. The most convenient definition is obtained by putting 



jo Mr. Ernest W. Brown, Theory of the 

so that 

,E+3fy /{m) (23) 

where/ (m) is a function of m which, in the case of the Moon, is very nearly 
unity. 

The definition must now be extended so as to cover the case when 
inequalities of any order are being considered. The general form of all the 
inequalities which have arguments of the form £ 2 * (21 even), are given by 

uf l =&%la i + 2(#'£' r 'V^"''^^" / « 2 '') i e 2p 'e' 2r 'k 2 «'a'- s ']C 2i 
where 

i—o, ±l, ±2, , . . ; p', q', r', s'=o, i, 2, . . , (except p'=q'=r' = s'=o) ; 

and £ ( )i denotes the sum of all such terms for the values of p', q f , r', s', 
given. The coefficient of £° in u%~\ by means of which a is to be defined, is 
therefore, since a =i, 

a[i + %(^'e v v r 'ri' r 'k'''k v a 2 '') e v e' v k v a 2s '], 
=a(i + >), suppose, 

so that i' is a small quantity of the second order at least. 

There are two practical methods of defining a, each of which has its use 
according to the equations we employ. 

One of these is to so define a that every term in v is zero, and therefore 
that the coefficient of £° is represented by a at every stage of the approxima- 
tions. This definition requires the determination of some additional terms to 
a whenever we are finding inequalities of the form Z? 1 . If we are using the 
homogeneous equations, this is undoubtedly the best definition, for then the 
determination of the additional terms in a can be left till the end of the work, 
and as a only appears in the parallax and not in the longitude and latitude, 
a very great degree of accuracy in its value is not required. In using the 
equations (3), (4), however, it would cause inconvenience as we should then 
have to find a further approximation to a at each step. As the latter equa- 
tions are those mainly used here, we shall adopt another definition better 
adapted to the calculations. 

This second definition (which we shall use below) is to give to a the 
meaning which it receives from the intermediate orbit only and to retain it 
throughout. Thus a is defined by the equation (23) making &{(E+ ]\[)/n 2 }~ h 



Motion of the Moon. 



71 



a numerical constant which never alters. The coefficients v are now no longer 
zero, but are definite functions of e 2 , e n , k 2 , a 2 , being determined along with 
the other coefficients of £ 2 ' in the ordinary way. This definition is necessary 
because of the want of homogeneity of the equations (3), (4).* In finding 
the parallax from the value of u, all that will be necessary will be to find a/r 
and then to multiply all the terms by a numerical quantity (which approaches 
unity very closely) in order to obtain [{E+ M)/n 2 f/r— the quantity usually 
obtained by lunar theorists. 

Hence, the linear constant a is defined to be the coefficient of £° in w £ -1 , where 
m £ -1 represents the intermediate orbit or variation curve only. Its value is given 
in Chap. II., and it retains this value throughout the whole of the approxi- 
mations. 

25. Definition of e. The first of equations (13) for the determination of 
the inequalities depending on the first power of e is, since A=o, 

£-\D + m)X + Mu.t 1 + Ns£= o, 

the solution of which is obtained by assuming 

M6 r i =ae2 i (^ +0 + <r 3i " c ), i=o, ±1, ±2, . . . 

When the substitution is made, and the coefficients of the various powers 
of l 2i±a equated to zero, we obtain a series of equations of condition for the 
determination of the unknowns e i5 e'*, c, which, are homogeneous and of the 
first degree with respect to e„ e' t . The determination of c is made by con- 
sidering the necessary relation which must exist between these equations ; it 
is actually found by a different method (see § 28 (b)), and we suppose it 
known. One of the ee t , ee'*, is arbitrary. The values of e t , e' t may all be 
made definite by taking e as the arbitrary constant and putting 

e n — e'n = I. 



* This is the definition intended in the remarks on p. 343 of the Investigations. A want 
of clearness in the statement of the definition in that paper has caused a misapprehension of 
its meaning. Mr. P. H. Cowell in his paper " On the Inelinational Terms " (Amer. Jour. 
Math. vol. xviii.) uses the homogeneous equations for the determination of the coefficients 
(kh') b and naturally finds it more convenient to put (k/c') =o. He, however, does not find 
the addition to the value of a (there called a ) ; this requires a reference to one of the 
equations containing k. The definition is, I hope, made quite clear in § 24 above. 



72 Mr. Ernest W. Brown, Theory of the 

The coefficients of £°, £-° in u(r l are aee , aee ', respectively. Since, by 
equations (9), 

u£- 1 =p cos (V— nt— t) + ip sin (V—nt — t), 

the coefficient of a sin I in p sin ( V-nt-e) will be * 

e( e o- £ o')= e ' 

by the use of the assumed relation. The value of e thus defined is very 
nearly twice the constant of eccentricity used by Delaunay. 

The general form of all the terms in uC 1 , which involve £ c , £~ c , is (§ 14), 

+ (ei'V^yV ''^''' a2< \U ' c ] e v e' 2 '"k V]. 

The definition of e must be extended so as to cover all these terms. It 
has been found most convenient to define it to be such that the coefficients 
of £ c and £~ e in the above expression are equal, except when p'=f = r'=s' = o, 
when we have already defined it by making their difference unity. Denot- 
ing, for a moment, each of these equal coefficients by /3, so that the terms 
containing £ c , £ _c in u^ are given by 

it is evident that the coefficient of a sin I in p sin ( V—nt-t) will be e(e — e ') 
=e. 

Hence the constant of eccentricity e is defined to be the coefficient of a sin I 
in the find!, expression of p sin (V— nt—e) as a sum of periodic terms, where 
V—nt—e is the difference of the true and mean longitudes and p is the projection 
of the Moon's radius vector on the plane of reference. 

26. Definition of k. The second of equations (13) for the determination 

of £ k is 

B%- 2 Mz u =o, 

in which we substitute 
where k' i — — k_ i . 

* This definition, for the terms with characteristic e, is the same as that which I adopted 
in " The Elliptic Inequalities " (Amer. Jour. Math. vol. xv. p. 261) ; e is there called Y t} . 



Motion of the Moon. y. 

The constant k is now defined to be such, that 

k = — A' =i, 

so that 2k is the coefficient of a sin F in the expression of z as a sum of 
periodic terms. The constant k differs little from Delaunay's constant y. 
The general form of all inequalities containing only £ ±g in zi is 



ak2„ . 



s ,( £ i>y "V y >'k l + "'k'"'a h \e^e ' 2r k 2 'V'(£« - £"«), 



and the definition of k must be extended so as to cover all these terms. 
AVe now define it to be such that 

(€*V*yy r 'k l + «l'«'a 2s ') = o, 

for all values of p', </, r, s\ except for p'=tf =)'={/ =o, when we have already 
put () equal to unity. It is to be remembered that if we interchan o-e the 
accents and the sign of i, the coefficient merely changes sign ; hence the co- 
efficient corresponding to that just written down is also zero. 

Hence, the constant of latitude k is defined to he the coefficient of 2a sin F 
in the expression of z as a sum of periodic terms. 



Section (yii).— Solution of the Equations of Conditio?!. Motions of the Perigee 

and Node. 

It will here be necessary to divide up the subject according as we are 
treating the equation (17) for u, the equation (18) for z, the homogeneous 
equations (20), (21) for u, or the homogeneous equation (22) for z. 

27. The Equation (17) for u. The general type of equation for the terms 
of characteristic X in u is 



£-\D + rnfu, + MuJT>- + A\£= a,U 



(24) 



where it is to be remembered that 31, N are known even-power series in £ 
with numerical coefficients (equations (14)) and A contains the known terms 
with characteristic X arising from the right-hand member. In only one case 
do the latter terms contain an unknown quantity ; this case, which involves 
the determination of a part of the motion of the perisree, will be treated in 
§ 28 (Jj). 

Royal Asthox 80c, Vol. LIII. 



m 



74 Mr. Ernest W. Brown, Theory of the 

Of the terms with characteristic X, suppose that we require to know those 
in w£ _1 which involve £ 2,±T , where t is one of the values of 

±pc±rm±2g , g, 

the right-hand member containing such terms. Let, therefore, these terms 

in A be denoted by 

A=% i (A i ^ + A'ff^). 

We substitute 

M J £- l =aXS i (A..£*"+*'<£*-')- ... (25) 

where \, \' ( are the unknown coefficients to be found, and equate to zero the 
coefficients of £ 2 ' +T , £~ 2i '~ T . The result is 

(2i + r+i+m)% + ^ j M J X i _ j + 2 J N J X' J . l =A i } 
(2i + T-i-mY\ , ^ + 2 j M J \'^.. J + 2jA T j\j +i =A'„ i ) 

where j=o, ±1, —2, . . . and 21 either =0, ±2, ±4, . . . or =±1, ±3, 
±5, . . . Since r multiplies X; and \'_ t it is evident that the complete 
values of c, g on the left may be replaced by their values c , g u which are 
functions of m only. 

The 3f ( , N f have quickly decreasing values for increasing values of i (see 
Chapter II.) and, in general, the same remark applies to A ( , A'_ t , \„ X/_ t -. 
The equations may thus be solved by continued approximation. The 
unknown terms of lowest order in the equations (26) are 

[(zi + T+ 1 +mf + M ]X i + F XL i , 

F \; 4 [(21 + r - 1 - m) 2 + M ]\' .; , 

respectively. The equations (26) are therefore those of principal importance 
in finding X f , \'_ f . 

When we solve them as two simultaneous equations to find \ i} \'_ u the 

common divisor is 

[(24 + T+ i+mf+M ] [(2t + T- 1 -m.f + M ]— ¥ " ; 

and, by the results contained in Part III. of the Investigations, it will be seen 
that this is very nearly equal to 

(2i + Tf[(2i + Tf-C Z ] (27) 



Motion of the Moon. 75 

If we had eliminated all the other unknowns from the equations (26), this 
expression would have been a factor of the divisor, the other factor being- 
very nearly unity. In considering the solution it is then only necessary to 
treat the cases where the expression (27) becomes small. 

28, There are four special cases to consider — namely, the cases when 
either factor of (27) is zero or small. We recall (§ 14) that when 
inequalities involving odd powers of a are under consideration, 21 is an odd 
positive or negative integer. It is unnecessary to prove many of the state- 
ments made below ; their truth, if not evident, can easily be demonstrated. 

(a) The case 2i+r=o. Here we must have t=o, i=o, owing to the 
incommensurability of c, g, m, 1. It is a question of determining coefficients 
of £ 2! ' ; there are no coefficients \\ and the two equations coalesce into one 
which is of principal importance in determining X . 

(b) The case 2i + t= ±c . Motion of the Perigee. Here i=o, r=c ; t= — c 
is the same case as t=c as we consider £ c , £~ c together. In this case A con- 
tains an unknown quantity — namely, the part of the motion of the perigee 
which has the characteristic X/e, say, c A/e , and it will be found possible to 
put 

where B t , b f , B' i} b\ are entirely known. It will be found also that b t , b \ are 
always the same whatever X may be. 

Substitute these values for A i} A' _ t in (26). Multiply the first equation 
by e ( , the second by e'._,-, and take the sum for all values of i. Since t=c , we 
find 

2l( 2 i + c + 1 + m) 2 ,V, + ( 2 i + c - 1 -m) 2 \'_ t e'-J 

+ VfiMAk^ + X'_ w£ '-i) + ZiVW-A + W-<) 

= 2 i [B i e i + B^^ + c x , e (b i c i + b'_^^)] (28) 

But for the terms with characteristic e, we have 

(21 + c + 1 + m)% + VjM fi ^ + XjF/j^=o, 
(2i + c — i- my_i + 2jMjt'_ M + ZjNfy, ,= o. 

Substituting these in the previous equation, it is easily seen* that the left- 
hand member vanishes and therefore the right-hand member of (28) equated 
to zero determines c A/e . (See Investigations, p. 336). 

* For we have 'SfiM i X i ^ J € i ='SfiM__Jk i e iH and M H =M p etc. 



76 Mr. Ernest W. Brown, Theory of the 

When the value of this quantity has been found, the equations (26) may 
be solved by continued approximation, all the \ { , \ f ( being expressed in terms 
of X , X'f, One of the two equations for \ , X' can then be deduced from 
the other. An arbitrary relation connects the X ; , \',. We have already 
(§25) settled this relation to be such that X =X' . The determination of a 
new part of the motion of the perigee thus goes with a more accurate defini- 
tion of the constant of eccentricity. 

The manner in which c may be best obtained is fully discussed by Dr. 
Hill in his paper " On the Part of the Motion of the Lunar Perigee, &c." 
(Acta Math. vol. viii.). Its value is there found to fifteen places of decimals. 
The parts of c which have the characteristics e 2 , e' 2 , k 2 , a 2 properly belong to 
Chapter V. (" Inequalities of the Third Order ") of this memoir. The calcula- 
tions, however, have been advanced in this particular direction so that c e , c e », 
c k * might be obtained ; they will be found in the appendix to Chapter IY. 

(c) The case 2i + r small compared with unity. The inequalities which are 
of hnq period compared with the lunar month. A troublesome defect of the 
method arises here. The divisor (27) contains the square of 21 + t, while the 
corresponding coefficients \, X'_, are in general of the same order of magnitude 
as A t /(2i+r). The reason of this is easily seen on solving the equations : 
one of them, in fact, generally differs from being deducible from the other by 
a quantity which is of the order of magnitude ii + t. This is illustrated in a 
striking manner by the long-period inequality whose argument is 2F-2Z 
(which is one of the most troublesome in any method). Here 2*'+T=2g -2c 
= +-0272, nearly, and therefore (2z' + t) 2 = + -00074, while the corresponding 
numerical coefficients are of the order of magnitude unity. 

The difficulty in all these cases is best avoided by computing the 
homogeneous equation (21) for the particular value of 2i+r, and combining it 
with one of the equations (26). In (2 1 ) the terms divided by (21 + r) 2 have the 
factor m 2 , and therefore with the same degree of accuracy in the known parts 
of the equation we are able to obtain X„ \'_,more accurately than if we simply 
used the two equations (26). 

(d) The case 21 + t±c„ small compared with unity. The numerous short- 
period inequalities the wean motions of whose arguments approximate to that of 
the principal elliptic term ■ e.g. the Evection and the Parallactic Inequality. The 
method is not in defect here, since the divisor is of the same numerical 



Motion of the Moon. 



77 



magnitude as that arising in any other method. It will be noticed that the 
divisors are smaller according as the periods approach more nearly to that of 
the principal elliptic term and not to the lunar mean sidereal or synodic 
periods. 

29. When the pair of equations for X;, k'_ t possess a small divisor, the 
approximations proceed slowly. In some of these cases it is advisable to 
save labour by finding X i±1 , \' t±1 , in terms of X,-, X'_ t and the known quantities, 
from the equations with suffixes ±i±i, and to substitute the results in the 
equations with suffix i before solving the latter. In all cases where the 
difficulty occurred, it has been avoided by this device. 

30. The Equation (18) for z. The course of the argument is very 
similar to that in §§ 27, 28, and therefore the results will be given briefly. 

The general type of equation for the terms with characteristic X in z is, 
by the equation (18), 

D\-2Mz^&XA l . ... (29) 

where A represents the known terms. For the terms which involve £ 2,±T , we 
find 

A=z t A l (£«--F"-') (30) 

and substitute 

^aXSAKi^'-r*-') ■ (31) 

since in z we always have X'_,- =— X,. The equations of condition are 

(2i + rfX i -2^,M j \ i _ i =A i (32) 



where j=o, ±1, ±2, . . . and 2/ either =0, ±2, ±4, . . . or=±i, ±3, ±5. . . . 
The equations (32) are solved by continued approximation — that written 
down being of principal importance in finding X,-. The coefficient of X, is 

(2i + Tf— 2i)I 

which, if we had eliminated all the other unknowns, would have been 



(2i + T)*-g * 



- (33) 



multiplied by a numerical factor which is nearly unity. 

31. There are only two cases to consider, namely, those in which 21 + t 
is zero or small compared with unity. Hence no long-period inequalities can 






78 Mr. Ernest W. Brown, Theory of the 

give rise directly to large coefficients in z ; thus the cases (a), (c) of § 28 do 

not arise ; the cases corresponding to (b), (d) are those numbered (//), (d') 

below. 

(//) The case 2i + r=±g . Motion of the Node. Here i=o, T=±g , and 

A contains an unknown part of the motion of the node of the form g A;k . 

We find 

A^JJt + g^bf, 

where B h b t are entirely known. Substituting for A ( m (32), multiplying the 
equation by ^ and summing for all values of i, we obtain 

H 2i + go) 2 * A - 2 ^jMjX^jJc^ Zjlfa + g^Xfifa. 

But for the terms with characteristic k we have 

(2i + g Q )%—22,M J k i _ J = o. 

On substituting this in the previous equation, the left-hand member 
vanishes and we find 

g vk =-(2,iB,£<)-s-<SM) (34) 

When this has been calculated, the equations may be solved by continued 
approximation. The existence of the relation (34) implies the arbitrariness 
of one of the \ : this has been defined to be such that \=o (§ 26). 

The determination of g has been made by Adams (M. N. vol. 
xxxviii., Coll. Works, vol. i.) and later by Mr. P. H. Cowell {Amer. Jour. 
Math. vol. xvlii.) where full explanations of the method used will be found. 
The advance numerical results for g e , g e «, g k =, are given in the Appendix to 
Chap. IV. below, g k * having been found by Mr. Cowell in the paper just 
referred to.* 

* Mr. Cowell objects to the above (which was given in a slightly different form in the 
Investigations) as a practical method, en account of the supposed length of the calculations. 
He, however, uses equation (8) to find g k » and the accompanying coefficients (P&')„ while 
the above method contemplates the use of the equation (4). In the latter, the coefficients 
A- or B t , b t are determined just as in any other set of inequalities ; the only labour that 
remains in order to find g x/k is the few minutes' work necessary to calculate the equation (34) 
above. The answer is, therefore, that the homogeneous equation (8), or the equation (4) 
should be used to calculate both the coefficients \ and g x;k , not one for the coefficients and the 
other for g Ak . 



Motion of the Moon. 



79 



(d') The case 2i + r±g small compared with unity. The inequalities 
whose periods are nearly equal to that of the principal term in latitude. The 
remarks made in § 28 (d) apply also here with one or two evident changes ; 
they need not, therefore, be repeated. 

32. The Homogeneous Equations (6), (7) for u. — Suppose that it be 
required to determine the terms in ul" 1 with characteristic X and arguments 
2i±r, where t is one of the values of ±pc±rm±2qg, those with lower 
characteristics having been found. It will be necessary here to slightly alter 
the notation of the last paragraphs by specifying the whole of the argument 
as well as the characteristic in the notation. 

Let the particular terms in u-X" 1 which have the arguments 2i±r be 
denoted by 

aXS^^' + X., ,(?-) ... (35) 



The coefficients \ Tii , X_ Tii are the unknowns, and are the same quantities as 
those previously denoted by X,-, X' ; . 

The equations being of the second degree with respect to u, s, z, we must 
consider how terms with characteristic X and arguments 2i±r arise in such 
expressions as D\us), u 2 , &c. The required terms will be made up of terms 
with characteristic ju,, arguments ±(22 + 0-), combined with terms with cha- 

(21 + t — cr), where 



racteristic v and arguments 



jJ.V = X. 



In conformity with the notation for terms with characteristic X, let these 
terms be expressed by 



ul~ l 



y^U t 'r-.,t^'-'+y.-r,iS u -' + ') ) 



(36) 



It is evident that these may be made to include the terms (35) by putting 
[m = \, ct=t in the first, or v=\, cr=o in the second. 

To obtain the corresponding terms in s, we put i/£ for £. As / receives 
negative as well as positive values, these may be written 









(36') 



So Mr. Ernest W. Brown, Theory of the 

Whence, for all terms with characteristic X and arguments 2J + t, 

8*=a 1, XS„y i ,/*_., tV_ r+ „, -j-t-J?**, etc. 



(37) 



Since j, e were supposed to have the same range of values, it is evident 
that i may receive the values o, ±i, ±2, . . . while 2/ receives either the 
values o, ±2, ±4, . . . or the values ±1, ±3, ±5, . . . according as X con- 
tains even or odd powers of a. It is not necessary to specify summation 
with regard to /u- or v ; it may be understood in the summation with regard 
to cr. The above expressions are so adjusted that the mme set of values for 
cr will be available in all of them. 

In general, for characteristics of orders which give sensible terms in the 
lunar theory, the number of values of yx, v, cr is quite small. [For example, 
if X=eV, T=2C—m, we have the following pairs of values for /a, <t respec- 
tively : 1, o ; e, c ; e', -m ; e 2 , 2c ; ee', c-m; eV, 2C-m. The corresponding 
values of v are derived from the relation pv=\.] When /x= 1, cr=o, the co- 
efficient is i(y ; it is that denoted previously by a t . 

The equations (6), (7) must now be put into the forms which will give 
results best adapted for numerical calculation when we substitute the expres- 
sions (37) in them. 

The first process is the calculation of the terms involving z, Q 1 in the 
equations ; they are evidently known terms, and the calculations consisting 
chiefly of multiplications, the latter do not call for special remark. 

When these terms have been obtained the equation may be put into the 
form, after integrating (7), 

L i (v.s)-DuI)s-2m(uDs-sDu) + 9 -m%u + sy=L (38) 

uDs-sDu-2mus + 3 m 2 D- l (u 2 — « 2 )=A (39) 

where L contains the known terms arising from z, & h in (6) and DA the 
known terms arising from 81 in (7). No arbitrary constant is needed in A, 
for the coefficient of £° always vanishes in (7). 



Motion, of the Moon. 



The forms of the terms which go to make up L, A show immediately 
that, for terms with characteristic X, and arguments 21 + t, we shall have 

A=a 2 ASA(r i+r +r 2; '~ r )- 

The left-hand members of (38), (39) are also of similar form ; hence, it is only 
necessary to equate the coefficients of £ 2J+T to zero in (38), (39) in order to 
find those of £ 2i±T in utr 1 . 

Multiply (39) by 2m + 1 and add to (38). The result may be written 



]JP(u8)-D(uZr x ) . D(s£)-(i +2m-^m 2 V 



(40) 



+ 9 m 2 M 2 + ^m 2 (2m+ OD-'^ + ^mV-J m 2 ( 2 m + i)/)" 1 ^) 

=i + (2m+i)A 
an equation which replaces (38). 

33. Substituting the results (37) in (40), (39), and equating the co- 
efficients of £ V+T to zero, we find 



(2J + T) 2 — (2i + <r)(2J — 2i + T — ar) -i — 2m H — m 2 /x„ ti v a _. 



, (a „ , % ,2m + i\ 
V4 2 2j+rJ 

+ ('9 m2 _3 m ^±l > \ "l =jZ: . + ( 2m + I )A ; 

\4 2 2J+T/ J 



X,i 



(2j-4i + r-2o— 2ra-2)fi <rii v,_ Ttt _ J - 
j. 3 m * f, 



2 2J + 



T I J , 



=A,- 



(4i) 



(42) 



The unknown quantities which are found by means of these equations 
are given by the values [x = \, ct=t, and /x= i, o-=o. Since the a t are known 
numerical quantities, the equations are linear with respect to the unknowns 
X ±r j and they can be solved by the ordinary methods of continued approxi- 
mation. 

The equations written down are those of principal importance in finding 
X ; X_ T _•. The principal terms involving these two quantities are obtained 
by putting 

cr=r, i=j ; whence /j,„ i =X r y and iv-t, i_,,—«o =I > 
cr=o, i=o ; whence // [ , £ =a =i and i',_ r ,,-_,■= X_,,_/- 

Royal Astron. Soc, Yol. LIII. k 



82 Mr. Ernest W. Brown, Theory of the 

The corresponding terms in the respective equations are 

[( 2 i + r) 2 -i- 2 m + Jm 2 }(X r ,, + X_ T ,__,), j 



2 ' ' I" (43) 

-( 2 j + T )(\ rii -X_, H ) - (2 + 21B)(\ ,, + X.. t , _j), j 



The method of solution of the simultaneous equations is therefore 
evident. We find the sum of the two unknowns from the first equation, and 
thence, substituting, their difference from the second equation. 

The greater part of the labour of calculation consists in the computation 
of the first term of each equation, owing to the fact that the coefficients of 
h<r,t v <T-r,i-j are different for different values of i,j. The coefficient in question 
in equation (41) is best written 

(t 2 -ctt + ct 2 — 1 -2m + 1 lmA +{4f—4ij j- 41 2 ) + (4J—2i)r + (4i — 2J)o: 

The first term of this remains the same while cr remains the same and 
when r— cr is put for cr ; the second term is always integral ; the third and 
fourth terms require only multiplication by integers. Hence, after the first 
term has been obtained, no logarithmic multiplications are necessary to find 
the whole set of coefficients corresponding to the different values of i, j. The 
last remark applies also to the corresponding coefficient in (42). The rest 
of the numerical coefficients in both equations do not involve /. 

34. We can deduce from (41), (42) forms in which the coefficient of 
a \ Tj is— 1 and that of a X._ T> __ ; - is o, so that there will be no need to solve two 
simultaneous equations as the final step in each approximation to a pair of the 
unknowns. The details of the algebraical steps are sufficiently simple, and I 
merely give an outline.* 

The expressions (43) show that if we multiply (41), (42) by 



2m + 2 — 2J — r, ( 2J + t) 2 — i — 2 m + - m 2 , 

respectively; add and divide the resulting equation by 

2(2J + t)U2J + t)' 2 — i-2m + -m 2 |, 

* Further details will be found in Chap. XI. (ii) of my Treatise on the Lunar Theory, 
where the spirit of this method is applied to a particular ease. 



nm 



% 



Motion of the Moon. 



83 



the coefficient of A._ T> _, will be zero, and that of \ rJ will be — 1. The result is 



44) 



where 



2J + T, 2*4-0"] 



21 + a- 2(2/ 4- t) 2 — 2 — 4m 4-m 2 4- (2*4-0- — 2 j — r)(2j' + T — 2 — 2 m) ' 



2j4-r 



2(2; 4- r) 2 — 2 — 4m 4- m 2 



r 2 ?4-- 1= 3 m a (2i4-r) 2 -4( 2 i4-r)-2-2(2i4-r4-4)m-9m a 

lJ '' J 4 (2J4-r) 2 {2(2/4- T ) 2 -2-4m4-ni 2 } 

/ 2 ,- + r \_ 3 m 2 5( 2 i + '-) i '-8( 2 i4-T)4-2-2( io/4-5r-4) m4-9m 2 

W '' 4 (2i + r) 2 {2(2i + r) 2 -2-4in4-m 2 } 



.» (45) 



(21114-2 — 2J — t)Lj ,4- j(2 i /4-r) 2 — i — 2m4--m 2 4-(2m4-i)(2m4-2 — 2j — t) ;- A,- 



//,,;•=- 



(2j'4-t) {2(2J+-) 5 -2-4m4m 3 } 



... (46) 



The equation for V_ T> _,- is obtained from (44) by changing the signs of t, j. 

35. The case (0) of § 28 deserves mention in connection with equation 
(44). It will be noticed that whenever the square of 21 + t appears as a 
divisor, namely, in the second and third terms of (44) and in Aj/(2J+r), 
(since A ; already contains it as a divisor), the terms have the factor m 2 . The 
use of equation (7) or (42) for the calculation of the coefficients of long- 
period inequalities has been noticed in § 28(c). Thus (41), (42) and (44) 
are free from the objection remarked in that paragraph. 

It may be also noticed that the expressions (43) show that when 2J+t 
is small, the corresponding loss of accuracy in the results arises in the differ- 
ence of \ Tjl X_ Tj _; and not in their sum. This, translated to polar coordinates, 
is equivalent to saying that the loss of accuracy in the coefficients of loner- 
period inequalities is chiefly felt in the longitude and but little in the parallax. 
In the case of short-period inequalities with small divisors the loss of accuracy 
falls on both coordinates. See also § 31. 

With regard to the relative advantages and disadvantages of the equa- 
tions (3), (4) and the homogeneous equations, from the point of view of actual 
calculation, it seems, from- the experience gained in using both forms, that 
on the whole the advantage lies with the former. This is certainly true for 
inequalities of the second order and very probably for those of the third order 



84 Mr.. Ernest W. Brown, Theory of the 

also. For inequalities of higher orders, the operations in the former are 
numerous but simple, and capable of continual verification ; those in the 
latter are rather less numerous but more complicated, more productive of 
error, less easily verified at the various stages, and are much less easily 
arranged for a computer. The equations (3), (4) are now being used for the 
inequalities of the third order. 

If the homogeneous equations be used, the expansions (41), (42) are pre- 
ferable in actual calculation to (44). This remark does not, of course, apply 
to the inequalities of zero order, computed by Dr. Hill in the Researches. 

36. The Homogeneous Equation (8) or (8') for z may be treated in a similar 
manner, and a formula similar to (41) or (42) obtained for it. The calcula- 
tions and results are much simpler, first on account of the less complicated 
form of (8) ; and, secondly, on account of the fact that there is only one 
equation of principal importance for the determination of each unknown, 
instead of two equations for each pair of unknowns. As the equation (4) 
will probably be used for all inequalities, I shall not develope the equation (8). 

The motions of the perigee and node are found by continued approxima- 
tion along with the unknown coefficients. It is possible to eliminate all the 
latter and to get single equations for the parts of the motions of the perigee 
or node corresponding to those given in §§ 28, 31 above ; but this will be 
no saving of time, as the equations are somewhat complicated. The formula) 
necessary will therefore not be given here. 

Section (viii). Calculation and Verification. 

37. It has already been stated that one of the chief objects aimed at in 
the developments given above was the reduction of the calculations to forms 
which made them as far as possible merely mechanical. This is the case 
with the right-hand members of (17), (18) or (17'), (18'). In the case of the 

former, — - r , — 2 , are calculated, once for all, from the values of u ,s . 

At each stage of the approximations ujn is obtained from the value of u^ by 
multiplication of uj,' 1 by i/u^ l =s ii £,/p l ?, the last quantity being found by 
" special values-" The remainder of the calculations of the right-hand mem- 
bers are then simple multiplications of £ series. In the case of the equations 
(17'), (18') (which were used for the results of Chap. IV. and for some of 



Motion of the Moon. 



those which will be published in Chap. V.) P, P, Q, . , . were calculated 
by the method of special values from those of u , s ; the remainder of the 
process is then as before. 

38. The plan adopted for the multiplication of any two ^-series, say 

consists in taking out the logarithms of the a t and arranging them along a slip 
of paper in the order . . . a 2 , a 1} a , a_ u a_ 2 , . . . ; the logarithms of 
the /3; are arranged along another slip in the order . . . /3_ 2 , ft_ u fi , yS x , /3 2 . 
. . . The two slips being placed over one another, the sums of all the 
logarithms for a given value of; are taken without moving the slips, and they 
are written down in a column. The number corresponding to each logarithm 
is then taken from the tables and the results added for each value of j. Thus 
to find the coefficient of t, i+a+fi in the product, that is ^ a 2 _ s #, the slips are 
placed so that a 2 falls under /3 , then a x falls under /3 l5 &c. The arrangement 
of this part of the sheet is then — 



Values of 

3 i 
2 

— 2 

— 1 
o 



«V-;#i 



log 



number 



Sum, < /=2 



The process can be thus arranged for a computer, and the mere copying 
of figures from one sheet to another is very rarely necessary. 

The result of each multiplication of series is verified by adding the sums 
for all values of j. The sum should be equal to (X&D^Zfit). 

39. The values of the A t , A'_ { in equations (26), (32) are in general carried 
to the same number of places of decimals for each value of i. In the solution 
of the equations of condition, large divisors frequently occur for large values 
of i and the X ; , X',- are obtained to one or two more places of decimals for large 
values of i. Thus it is in general possible to find B"u A to the same degree of 
accuracy as the A„ A' { . Exception only occurs in the cases of some long 
and short period inequalities ; when it occurs, the corresponding values of 



86 Mr. Ernest "W. Brown, Theory of the 

A t , A'i are taken to one or two more places of decimals or the homogeneous 
equation (7) is used, as explained in § 28 (c). 

The object of taking the values of \„ \\ to more decimals for large 
values of i is to render the equation of verification more searching. For veri- 
fication I use one of the homogeneous equations with £= 1 ; the calculation 
of it is never very long, and it appears to furnish a good test. See Investiga- 
tions, p. 343. 

40. In the following chapters I give in general two sets of numerical 
results : First, the values of the right-hand coefficients A i} A'r, and, secondly, 
the values of X,-, X' t . They are taken exactly as they stood in my manuscripts. 
Although many of them will not be more than two units wrong in the last 
place given, the number of the calculations prevents this being said of all. 
They are intended to be trustworthy as far as the last figure but one in each 
case. The sums of the numbers in each column are always given, so that any 
error of transcription or typography may be detected should it occur. 

All calculations are made at least twice, separated by an interval of 
time. The general plan, when several hours a day were available, was to 
have two or three separate sets of calculations proceeding together. Each of 
these would be carried to a certain stage and, after the lapse of a day or two, 
they would be gone over again, the errors, if any, corrected, the results 
verified (whenever this was possible) and they would then be taken a stage 
further. In this way an error running through several pages of calculations 
was avoided. 

Section (ix). Transformation to Polar Coordinates, 

41. We have, by equations (9), Section (iv), 

p exp. (V — nt — e)t,=ul~ l , 
p exp.—(V—nt — e)i=s£, 
p tan <f>=z. 

Hence 

2i(F— nt — e)=log tit,' 1 — log sg, 



r (us + z 2 )* 
^tan" 1 -. 



Then 



and 



Motion of the Moon. 87 

Let V be the part of V corresponding to the values n , s of u, s and let 

2lV =log U (~ l — log S$(, 

2 t (V + SVJ=log (wo + Sw^r'-log (s + 2s„K> 
2l2 V,=lo g ( I+ ^)-lo g ( I+ ^) . 

t *0 2\Mo/ 2\«o/ 



Also 






tan y _.*%£ 1 - s o4_S,;(a,— a_j)sin 2iD 
t M { _1 + S ^ 2i(«i + «_,-) cos 2iD 



(47) 



(48) 



The value of V may be calculated by the method of special values from 
those of a { . The various terms of the right-hand member of (47) will have 
been already found in the calculation of the inequalities. Whence by addi- 
tion we obtain the true longitude V. 

42. With the same substitutions, we have 



2^V1 " 5 






(49) 



where 



iW-E+M. 



The right-hand member of this equation is then expanded. The various 
products will have been found, as before. All that remains to find a/r is to 
multiply the result by 



a a a 

r a ' p 



(since r =p„), 



which is found from Chap. IT. 



88 Mr. Ernest W. Bkowx, Theory of the 

We may also use equation (5) of § 7 to find i/r, the constant term 
being obtained by (49) or by the method contained in Part ii. of the Investi- 
gations. 



43. Finally 



-hc-mc-)'- 



and 



P Po 



P 3^ 



' 2w„ 2s„ 2«^ SSj." 
I ~j-~ — \~ — r . 

w s o w o s o . 



(So) 



The expansions and multiplications will have all been performed, and thence 

</> may be easily found. 

Hence the whole process of transforming to polar coordinates will be 
first, for the longitude, the addition of certain known series and the calcula- 
tion of V ; secondly, for the parallax, the addition of known series and a 
multiplication of the whole by a/p ; thirdly, for the latitude, the addition of 
known series. 



CHAPTER II 

Tekms of Order Zero. 

Section (i). Values of a { , a. 

44. The coefficients of order zero have been obtained by Dr. Hill * to 
15 places of decimals. They are given by the particular solution of equa- 
tions (12) or (12') of § 18. This solution is expressed by 

Mo ^" 1 =a2 i a i i' 2i , where a =i. 

The value of m = ri/(n— n') used is 

m= + -08084 89338 0831 1 6. 

* "Researches in the Lunar Theory," Aram: Jour. Math., vol. i. pp. 247-249. The 
coefficient denoted above by aa, is denoted by a, in Dr. Hill's paper. 



Motion of the Moon. 

Values of 



89 



i. 


a%. 


6 


+ "ooooo 


ooooo 


00007 


5 


+ "OOOOO 


ooooo 


01107 


4 


+ - ooooo 


0000 1 


75268 


3 


+ 'ooooo 


00300 


31632 


2 


+ 'OOOOO 


58786 


56578 


1 


+ "00151 


57074 


79563 





4-i 






— 1 


— -00869 


57469 


61540 


— 2 


4- 'ooooo 


01637 


90486 


-3 


4- 'ooooo 


00024 


60393 


-4 


4- 'ooooo 


ooooo 


12284 


-s 


+ "ooooo 


ooooo 


00064 


-6 


4- "ooooo 


ooooo 


ooooo 


Sum 


4- "99282 


60356 


45842 



The relation between a and a where 

n 3 a s =E+M. 



is given by 



^=4- -99909 31419 75298. 



This relation will not be required after the end of this Chapter until we 
come to the deduction of the lunar parallax and the expression of the co- 
efficients containing a=a/a / in terms of a /a'. 



Section (ii). — Values of M, N, P, etc. 
45. From the results of the previous section we deduce the series for 
M, JV, P, Q, S, S, T, 



a a ku c, k 
u ot' 1 ' Pq ap 3 ' ap/ 



See Chap. I., §§ 19, 20, Z7- The series for P, Q, ... are obtained from 
those for P, Q, . . . by putting i/£ for £; this is the same as putting -?' 
for i in the suffixes of the coefficients. 

Royal Astron. Soc, Vol. LIII. o 



go 



Mr. Ernest W. Brown, Theory of the 



Values of 



i. 


Mi. 


Ni. 


s 


+ "OOOOO ooooo 


5 


+ 'OOOOO 


00009 


4 


+ "ooooo 00056 


5 


+ 'OOOOO 


00824 6 


3 


+ •00000 06029 


7 


+ 'OOOOO 


70129 7 


2 


+ '00006 28883 


4 


+ '00054 


79401 6 


I 


+ •00630 84231 


2 


+ -03686 


55i7i 8 


O 


+ •58902 22856 


4 


+ 175707 


88032 7 


— I 


+ •00630 84231 


2 


+ -01078 


63527 2 


— 2 


+ •00006 28883 


4 


+ 'OOOOI 


25690 4 


-3 


+ •00000 06029 


7 


+ 'OOOOO 


00982 3 


-4 


+ •00000 00056 


5 


+ 'OOOOO 


00007 6 


-5 


+ 'OOOOO ooooo 


5 






Sum 


+ •60176 61259 





+ 1-80529 


83776 9 



The series M is given by Dr. Hill in his paper just referred to. The 

series M, N are both given on p. 328 of the Investigations. 

p=%p i ^ i , p=%p^ u , g=s&£*.* 

Values of 



i. 


• P». 


Qi- 


5 


+ -ooooo 


00005 


+ 'OOOOO 


00020 


4 


+ -ooooo 


00465 


+ -ooooo 


01462 


3 


+ '00000 


40164 


+ 'ooooi 


04704 


2 


+ '00032 


38766 


+ '00066 


73632 


1 


+ '02280 


40093 


+ -03476 


15314 





+ 1-17156 


77322 


+ 1-17132 


34260 


— 1 


+ '01 084 


18484 


— "001 1 2 


12092 


— 2 


+ '000 10 


24640 


+ -ooooo 


3 J 9 2 3 


-3 


+ 'OOOOO 


09526 


+ -ooooo 


00327 


—4 


+ 'OOOOO 


00092 


+ -ooooo 


00007 


-5 


+ 'OOOOO 


OOOOI 






Sum 


+ 1-20564 


49558 , 


+ 1-20564 


49557 



The series Q will not be required. 



Motion of the Moon. 



9i 






Values of 






i. 


Hi. 




*. 




Ti, 




s 


+ -00000 


00016 


+ -ooooo 


00004 


+ -ooooo 


00047 


4 


+ 'OOOOO 


01268 


+ -ooooo 


00384 


+ -ooooo 


03170 


3 


+ 'OOOOO 


93162 


+ -ooooo 


34249 


+ -ooooi 


96595 


2 


+ -00061 


°55 2 ° 


+ -00028 


51417 


+ -00105 


79688 


1 


+ -03299 


09904 


+ -02103 


0383! 


+ "04494 


46487 





+ I-I7IS9 


65 6 55 


+ 1-17171 


87304 


+ 1-17123 


01076 


— 1 


+ -00906 


65690 


+ -02103 


0383 1 


— -00289 


67092 


— 2 


+ -00008 


18396 


+ -00028 


5*417 


+ -ooooo 


06964 


-3 


+ -ooooo 


o7394 


+ -ooooo 


34249 


+ -ooooo 


00136 


-4 


+ -ooooo 


00066 


+ -ooooo 


00384 


+ -ooooo 


00006 


-5 


+ -ooooo 


0000 1 


+ -ooooo 


00004 






Sum... 


+ 1-21435 


67072 


+ i'2i435 


67074 


+ i'2i435 


67077 



The series P, Q are given on p. 328 of the Investigations. The co- 
efficient Ri of that memoir is here called i?_,-. The coefficients <Q 3 , Q_ 3 
are corrected here, each of them having been diminished by one unit in the 
ninth place. 

46. Values of the coefficients of £ 2i in the expansions of 



i. 


a 


1' 


a 
Po" 


5 






+ 


•ooooo ooooo 


I 


4 






+ 


•ooooo 00012 


2 


3 


— -ooooo 


00156 


+ 


■ooooo 01632 


9 


2 


— -ooooo 


35 8l 5 


+ 


•00002 29OO7 





1 


- -00151 


57497 


+ 


•00358 99818 


9 





+ '99997 


36392 


+ 


•99999 97077 


7 


— 1 


+ -00869 


54035 


+ 


•00358 99818 


9 


— 2 


+ -00007 


54483 


+ 


•00002 29007 





-3 


+ -ooooo 


06521 


+ 


•ooooo 01632 


9 


-4 


+ -ooooo 


00056 


+ 


•ooooo 00012 


2 


-5 






+ 


'OOOOO ooooo 


I 


j Sum 


+ 1-00722 


58019 


+ ] 


•00722 58019 


9 



92 Mr. Ernest W. Brown, Theory of the 

Values of the coefficients of I 21 in the expansions of 



i. 


apo 3 ' 








5 


+ 'OOOOO OOOOI 


3 


+ -ooooo 00000" 


4 


4 


+ 'ooooo 00144 


3 


+ 'ooooo 00048 


5 


3 


+ 'ooooo 15058 


3 


+ 'ooooo 05751 


9 


2 ' 


+ '00015 J 7769 


1 


+ "00006 875 1 1 


8 


I 


+ -01439 I 49° l 


8 


+ '00841 09322 


7 


O 


+ 1-17141 74324 


9 


+ 1-17144 79701 


8 


— I 


+ -00242 99016 


5 


+ '00841 09322 


7 


— 2 


+ -ooooi 62575 


2 


+ '00006 875 1 1 


8 


-3 


4- -ooooo 01171 


8 


+ 'ooooo 05751 


9 


-4 


+ 'ooooo 00008 


8 


+ -ooooo 00048 


5 


-s 


+ 'ooooo ooooo 


1 


+ -ooooo ooooo 


4 


Sum 


4-1-18840 84972 


1 


+ 1-18840 84972 


4 



These four series will not be required until Chap. V., as the calculations 
'O'f Chap. IV. were made with P, P, Q; for a few of the inequalities of 
Chap. V. the series B, R, S, T have been used. 

CHAPTER III 

Terms of the First Order. 

47. The terras of the first order have been treated in Chap. L, § 19. 
The results contained in this chapter are classified in the following table : — 



Section. 


Charaoteribtic. 


Arguments. 


Quantities found here. 


Coefficients. 


Motions of Args. 


(i) 

(ii) 
(iii) 

(iv) 


e 

e' 
a 
k 


2J±C 
2i±m 

±(2i + g) 


i?i. V'i 


Co 

go 



where 






'•*> —3s —5 



Motion of the Moon. 



93 



The formula) furnished by equations (15), (i 6) of Chap. I. will be given 
in each case, although the results may have been otherwise obtained. Refer- 
ences will be made to all previously published results. 

Section (i). — Characteristic e. Value of c . 

48. The value of c , which is the part of c depending only on m, has 
been found by Dr. Hill * to 1 5 places of decimals. It is, 

c = + ro7i58 32774 16012. 
The equation satisfied by c and the terms with characteristic e is 



The'solution is 



tt a £- 1 =aeS,(e J 4 !U+ « + e',?»- c ). 



The equations of condition for the unknowns having been obtained by 
substituting the assumed solution in the differential equation, and equating 
the coefficients of the various powers of £ to zero, we may solve them with 
the above value of c, so as to give e ; , e' s - in terms of e , e'„. 

Let 

£ »=Vo + A e 'o> e';=b' 4 « + /3y o . 

Values of 



i. 


h- 


ft- 


5 










4 


+ 'OOOOO 


00005 


— 'OOOOO 


00006 


3 


+ 'OOOOO 


00843 


— 'OOOOO 


00708 


2 


+ '0000 1 


47376 


— 'OOOOO 


85373 


1 


+ '00308 


02927 


— '00092 


80067 





+ 1 









— 1 


+ '01999 


88763 


+ '20567 


90112 


— 2 


+ 'OOOOI 


15205 


+ '00007 


34691 


-3 


■ — 'OOOOO 


00193 


— 'OOOOO 


01734 


-4 1 


— 'OOOOO 


OOOOI 


— 'OOOOO 


00012 


-5 










Sum ... 1 


+ I'023IO 


54925 


+ '20481 


56898 



' Motion of the Perigee, etc.," Acta Math. vol. viii. p. 35. 



94 



Mr. Ernest W. Brown, Theory of the 

Values of 



i. 


w. 


ft'. 


5 






— -ooooo 


00002 


4 


— "OOOOO 


00029 


— -ooooo 


00212 


3 


— 'OOOOO 


04039 


— -ooooo 


29218 


2 


— -00005 


93876 


- -00043 


20782 


I 


- -01054 


68058 


— '°7779 


5543° 










+ 1 




— I 


— -00108 


65960 


— -00019 


59999 


— 2 


-f -ooooo 


01043 


— -ooooo 


08618 


-3 


+ 'OOOOO 


00024 


— -ooooo 


oooS5 


-4 










-5 










Sum 


— -01169 


3089s 


+ -9 2I 57 


25684 



The arbitrary constant e is defined (Chap. I., § 25) to be such that 



«n~«n =1- 



Either of the two remaining equations of condition (those of principal 

importance for finding e„, e' ) then gives 

e 4-c' = — -49679 18022. 

From these two equations we find e , e' 0) and thence, from the numbers 

just given, the values of e h e\. 

Values of 



i. 




e i« 




1 


5 








+ 


•ooooo 


0000 1 


4 


+ 


■ooooo 


00005 


+ 


•ooooo 


00152 


3 


+ 


•ooooo 


00742 


+ 


•ooooo 


20851 


2 


+ 


•0000 1 


C0977 


+ 


•00030 


84234 


1 


+ 


•00146 


95307 


+ 


•05556 


82459 





+ 


•25160 


40989 


— 


•74839 


59011 


— 1 


— 


•14889 


75297 


— 


•OOOI2 


67065 


— 2 


- 


•00005 


20854 


+ 


•00000 


06713 


-3 


+ 


•ooooo 


01250 


+ 


■ooooo 


00048 


-4 

-5 


+ 


•ooooo 


00009 








Sum 


+ 


■10413 


43128 


— 


•69264 


31618 



Motion of the Moon. 



95 



I obtained these results by the use of the homogeneous equations.* 
A different set of values for b t , V u /3„ I3\ will naturally arise if we use the 
equation at the beginning of this section. One slight error which occurred in 
the reduction of b' 4 e + y8' 4 eo to the final value of e' 4 was discovered and corrected. 

The short-period inequality with a small divisor is the "Evection"; 
the corresponding coefficients are e_ x , e\. 

Section (ii). — Characteristic e'. 

49. The equation is 

Z-\D + mfu e , + Mu/C l + Ns/(= - ȣ-' H~\ 

os 

In the right-hand member we put s 1 =&> 2 , z=o, u=u , s=s , and neglect 

powers of d above the first (Chap. I., § 19). 

Hence, by Chap. L, Sect, (iii), 



ds 






2*0 +" - fr 2 M 
2 



(- 1 



4 



The solution is 

The equations of condition are formed and then solved by continued 
approximation. 



Values of 



i. 


? 


Vi- 






Vi'- 




5 


— - ooooo 


ooooo 


°3 


+ 'OOOOO 


ooooo 


24 


4 


— 'OOOOO 


00004 


40 


-f- 'OOOOO 


00030 


59 


3 


— 'OOOOO 


00572 


63 


+ 'OOOOO 


03956 


99 


2 


— '00000 


76025 


41 


+ '00005 


22794 


42 


r 


— •00103 


48418 


2 


+•00695 


08210 


5 





— •09186 


93227 




+ •09869 


89451 




— 1 


—•03636 


42746 


8 


4- "00448 


82585 


5 


— 2 


+ 'OOOOO 


17438 


21 


— "OOOOO 


01475 


00 


-3 


+ - O0000 


00322 


23 


— 'OOOOO 


00041 


9 6 


-4 


+ 'ooooo 


00002 


08 


— 'OOOOO 


ooooo 


29 


-5 


+ 'OOOOO 


ooooo 


01 








Sum 


— •12927 


43 2 3 2 




+ •11019 


°55 12 





* " The Elliptic Inequalities in the Lunar Theory," Amer. Jour. Math. vol. xv. pp. 259-261. 



9 6 



Mr. Ernest W. Brown, Theory of tin 



The above method was used to calculate all these coefficients. The long 
period inequality with a small divisor is the " Annual Equation," having the 
coefficients i^, i/ - The method of § 29 was used in the approximations to 
these two coefficients. 

The values of the corresponding terms in the true longitude have been 
given in a note in the Monthly Notices, vol. liv. p. 471. 

Section (iii). Characteristic a=a/V. 

50. The equation is 

t l (D + m) V + Mu£-* + Ns£= - -j^-'- 1 ] ■ 
x ' as 

In the right-hand member we put £3 1 =<y 3 , z=o, </=o, u=u , s=s . Hence, 
by Chap. I., Sect, (iii), 

as 4 a 2 |_2 2 J 



The solution is 
where 



2 i=±i, ±3, ±5 
Values of 



22. 


(«)«• 


9 
7 
S 
3 
1 
— 1 
-3 

1 ~s 

I ~ 7 
I ~ 9 


+ •00000 OOOOI 
+ •00000 00072 
+ - ooooo 04839 

— -00005 88448 

-•06417 03547 

+ •17899 19628 
— '00293 82096 

— •00000 18325 j 

— 'ooooo 00029 ! 


• Sum 


+ •11182 32095 



Motion of the Moon. 97 

These coefficients I found to seven places of decimals in a paper " On the 
Parallactic Inequalities in the Lunar Theory " * by the use of the homogeneous 
equations. They have been recalculated and extended to ten places by the 
above method ; errors of one unit only in the sixth places of decimals in the 
values of a_ h a, were detected. Dr. Hill in his paper " On the Periodic 
Solution, &c.,"f using my former values as a first approximation has also 
recalculated these terms to a high degree of accuracy by a totally different 
method. 

The short-period inequality with a small divisor is the " Parallactic 
Inequality," having the coefficients a v a„j. 



Section (iv). Characteristic k. Value of g . 

51. The part of the value of g which depends on m only, namely g , 
has been obtained by Professor J. C. Adams and Mr. P. H. Cowell (see 
the references in § 31). The latter finds 

g = 1-085 17 14265 58. 

The slightly different result obtained by Adams is due to the use of a different 
value for m. 

The equation giving g and the terms with characteristic k is 

DH k — 2Mz k =o. 
The solution is 

The constant k is defined (Chap. I., § 26) to be such that 

k =i. 



* Amer. Jour. Math. vol. xiv. p. 157. A different notation is there used. 

f Astron. Jour. vol. xv. pp. 137-143. Dr. Hill informs me that the large correction 
which he obtained to my value of the coefficient of the Parallactic Inequality n longitude, 
amounting to 5 units in the fifth place of decimals in the value of a h — a_ s , was due to a slight 
error in reducing them to his form. As the resulting value was only used as a first approxi- 
mation, his final results are, of course, correct. 

Royal Astron. Soc, Vol. LIII. p 



98 



Mr. Ernest W. Brown, Theory of the 



Mr. Cowell finds the following values (be. cit. p. 113) 



o 
Values of 



i. 


*,. 


5 


4- -ooooo 


ooooo 


01 


4 


+ -ooooo 


0000 1 


75 


3 


+ 'OOOOO 


00299 


82 


2 


+ 'OOOOO 


58673 


61 


1 


+ -00151 


22192 


28 





+ 1 






— 1 


— -03698 


393*3 


94 ■ 


— 2 


— -00004 


65750 


01 1 


-3 


— -ooooo 


01755 


37 i 


-4 


— -ooooo 


00008 


87 


-5 


— -ooooo 


ooooo 


°5 


Sum 


+ -96448 


74339 


23 



I have verified these results by means of the homogeneous equation (7) 
with a 1=0, by putting £= + 1, — 1, successively, after the substitution of the 
values in the equation. 

The short-period term with a small divisor is that having the numerical 

coefficient k_ x . 



CHAPTER IV 

Teems oe the Second Ordek. 
Section (i). Formula. 

52. The general type of the equation for u for terms with characteristic 
X, and arguments 21 ±r, is, by Chap. I., § 20, 

l-\D-{-jnfu > .+ Mu x t,- 1 + NsJ:=^\A ... . . ... (1) 



where 
and 



A=ti (A ( F^+A't C M - r ) when.r^o, 
yl=Sj A i l 2i when t=o. 



Motion of the Moon. 
Here, by Chap. I., equation (i7')> 

aX4=Part, char c . X, in 



99 



OS 



+| p (s», n* + - 8 S - e (% o 2 + | ^ (s«j (so-3 * (%)' 2 ] - (2) 



5« — s k . 



In all cases s is derived from u by putting i/£ for £. The first term of 
(17') contributes nothing to the terms of the second order. 

Also, by Chap. L, Sect, (iii), substituting for u, s, z and neglecting 
powers and products of u^ s^, z^ we have 



i (* + SsJ a 2 + _(« + 2« 



,)&*] 



a 



^ (s 2 +2 So S*„) «3 +| (m 2 + 2 M 2m J C 3 +^ (t6 S + «0 2s„ + 



«o 2«> 3 j 



+3 



T3S 



3 J. IS 

16 



a' 2 |_i6 16 -* 



5 - "o 2 *0 



(3) 



where 



n 2 , 6 2 take their values as far as e' 2 -when multiplied by w 0) s ; 
a 2 , 6 2 take their values as far as e' when multiplied by u M s^ ; 
a 3 , C 3) C 3 take their values as far as e' when multiplied by u , s ; 
a 3 , c 3 , C 3 are unity when multiplied by u^ s^. 

The solution is of the form 

^(T^aXSAP'^+A',?'-'), when r^o, 



and 



u£- l =aX&M*, 



when t=o 



(4) 
(4') 



The process consists in first finding the series for A in each case, and 
then, after the substitution of the solution in the differential equation, to 
form the equations of condition for the unknown coefficients. These are 
solved by continued approximation, there being one pair of equations of 
principal importance in finding each pair of coefficients \, \'_ t . The known 
values c , go of c, g, and the definitions of e, k given in Chap. III. are suffi- 
cient. The further definition of the linear constant (§ 24) occurs in section? 
(ii), (iv), (v), (viii) below. 



ioo Mr. Ernest W. Bkown, Theory of the 

53. The general type of the equation for z, for terms with characteristic 
X and arguments 2% + t, is, by Chap. I., § 20, 

D % z x — 2Mz x =aXAi (5) 

where 

jl=Vi(i* +T -£" M -*)- 

Here, by Chap. L, equation (18'), 

aXJ.t=Part, char . A, in —^ 

\_ 2 OZ 

'(PSu^ + PSsJ^ (6) 



2 a 



Sw^ssMe + Mj. + M., '%Z l ==Z k . 

The first term of equation (18') contributes nothing to the terms of the 
second order. 

Also, by Chap. I., Sect, (iii), substituting for u, s, z the value? just given 
and neglecting powers and products of u^, s^, z M 



2 oz 2 al 



- l ^=^A + ^^H + s ) (7) 



-where b s takes its value as far as (/. 
The solution is 

s^aAS^p+'-C-*-') (8) 

The process is the same as before. There is only one equation of 
principal importance in finding any coefficient \ t . The known values c , g 
of c, g and the definitions of the arbitrary constants given in Chap. III. are 
sufficient. 

54. The following table gives the various classes of terms of the second 
order, with the sections in which they are considered below. 



Motion of the Moon, 



101 



where 



Section. 


K. 


Arguments. 


Type of Coefficients. 


(H) 


6 3 


2i±2C, 2i 


(' 2 ). (*'% («') 


(iii) 


ee' 


2i±(c+m), 2i±(c — m) 


(«»)> («V). («?'). (*'v) 


(iv) 


e' 2 


2i±2IIl, 2% 


(V% (i% (rrf) 


(v) 


k 2 


2i±2g, 2t 


(*»), (A' 2 ), (*#) 


(vi) 


ea 


2i!±C 


(ea), (c'a) 


(vii) 


e'a 


2%±m 


(r?a), (rj'a) 


(viii) 


2 

a 


21 


(a 2 ) 


(ix) 


ke 


±(2i + g + C ), ±(2»+g-c) 


±(Ac), ±(/5*') 


w 


ke' 


±(2i+g + m), +(2i' + g-m) 


±(%), ±(fo)') 


(xi) 


ka 


±(2%+g) 


±.(ka) 



-.2, 3:4 



«i=±i, ±3, ±5 



The coefficients in the last column have the suffixes i or i h when the 
corresponding arguments have them, that is, according as the coefficients do 
not or do contain the first power of a. It will be remembered that when 
the characteristic contains an odd power of k (that is, for the terms in z), 
(&Y)j=-(£e)_ 8 ., (^e) < =-(^e / )_ i , etc. Hence, Sections (ii)-(viii) contain 
all second order terms in u, Sections (ix)-(xi) all second order terms 
in z. 

The results selected for publication are the numerical values of A i; A' t 
and those of the unknowns X i} X',.. 

The degree of accuracy to which the various results have been carried 
depend, first, on the general numerical magnitude of the characteristic ; 
secondly, for A t , A\, on the cases where 21±t, 2i±r±c, or 2i± T ±s become 
small ; thirdly, for \, X' t , on the accuracy required for certain terms of hio-her 
orders. The approximate numerical magnitudes of the characteristics* are 
given by 



•xi, 



' = •017, k=-o45, a=-oo26. 



For further remarks on the numerical results, see Chap. I., Sect. (viii). 
References to previously published results are given in all cases. 



102 



Mr. Ernest W. Brown, Theory of the 



Section (ii). Characteristic e 2 . 

55. Here \=e 2 , and 21 + t has the values 2«'±2c forming one set of 
equations of condition and the value 2% forming another set. The values of 
A corresponding to the two sets are obtained from equation (2) of this 
chapter, and they are given in the following table :— 

Values of A. Coefficients of 



i. 


£2i + 20. 


f«-°- c . 


T 2i - 




5 


4- -ooooo 


0002 


4- -ooooo 


0397 


4- -ooooo 


0059 


4 


4- 'OOOOO 


0146 


4- -00002 


4815 


4- -ooooo 


4028 


3 


4- -ooooi 


0222 


4- -00125 


4227 


4- -00025 


3358 


2 


4- '00064 


6617 


+ •03959 


7603 


+ •01291 


0151 


1 


+ -03321 


1631 


- -08783 


1650 


+ •41545 


8343 





4-1-08874 


2 S37 


4--2ig26 


5334 


-+6585 


5712 


— 1 


— 'i°743 


6306 


+ -00383 


3333 


4- -08798 


3630 


— 2 


4- -00888 


°S39 


4. -00005 


3547 


4- -00151 


7439 


-3 


+ -00015 


0316 


4- -ooooo 


0664 


4- -OO0O2 


1196 


-4 


4- -ooooo 


2104 


4- -ooooo 


0008 


-(- -ooooo 


0263 


-s 


4- -ooooo 


0026 






j 4- -ooooo 


0003 


Sum 


4-1-02420 


7834 


4--i76i9 


8278 


1 4--05229 


2758 



The solution is expressed by (Chap. I., Sect, (iv)), 



vS-^arfs, W +a '+ W"*+ ("V 



Solving the two sets of equations of condition— namely, that giving 
( e 2\ (Y2), j^nd that giving (ee')* — we obtain the values of these coefficients. 



Motion of the Moon. 
Values of 



103 





(« 


2 >- 


<« 


,2 )*. 


(««% 


5 


+ 'OOOOO 


00000 1 


+ 'OOOOO 


00049 


+ 'OOOOO 


00004 ° 


4 


+ "OOOOO 


0001 I 


+ 'OOOOO 


04893 


+ 'OOOOO 


00459 5 » j 


3 


+ 'OOOOO 


01135 4 


+ '00004 


84244 


+ 'OOOOO 


47226 4 


2 


+ - OOO0 1 


16070 9 


+ '00428 


5788 


+ '00046 


03442 


1 


+ •00112 


37013 


+ •01564 


7028 


+•03917 


99373 





+ '09402 


3537 


+ •03180 


1697 


— -I 33" 


2689 


— 1 


—•06517 


3271 


+ '00006 


45654 


+ '01492 


2756 


— 2 


+ ■00133 


0056 


+ 'OOOOO 


06650 


+ '00002 


21364 


-3 


+ 'OOOOO 


T7404 


+ 'OOOOO 


00057 4 


+ 'OOOOO 


02603 


-4 


+ 'OOOOO 


00260 


+ 'OOOOO 


00000 6 


+ 'OOOOO 


00022 8 


-5 


+ 'OOOOO 


00003 






+ "OOOOO 


00000 2 


Sum 


+ •03131 


75 12 


+ •05184 


8668 


-■07852 


2483 



These coefficients were given to eight places of decimals on pp. 325, 323 
of my paper referred to in § 48, having been obtained by the use of the homo- 
geneous equations. The notation is different. The symbols (e*), (e' 2 ) are 
there denoted by f/Y 3 ,f/Y 3 respectively ; the symbol (ee'),- used here is not 
the exact equivalent of 8a,/ F 2 in that paper, owing to the meaning there 
assigned to a being different from that of a. To compare them we must 
put 

(«');=(«„ Stti+aj Sa )+1 7 2 a 2 , 

the terms in the right-hand member being the quantities contained in the 
paper referred to. 

An error of one unit in the sixth place of the value of (e 2 )_ 2) or/_ 2 in 
the paper, was discovered, inducing smaller errors in the other coefficients, 
All the coefficients have been re-calculated by the method of this memoir and 
the results, as seen above, extended to nine places of decimals. 

The long-period inequality with a small divisor is that having the co- 
efficients (e' 2 )_i, (e /2 )i- This was separately calculated by the homogeneous 
equations as in the " Elliptic Inequalities " and the results were verified by 
the equations of condition which the above method furnishes. 



104 ^ r - Ernest W. Brown, Theory of the 

Section (iii). Characteristic ec / . 

56. Here 2^+ T has the two sets of values 2^±(c + m) and 2Z±(c-m). 
The corresponding values of A are given in the following tables. 

; ; : Value of A. Coefficients of 



S 
4 

3 

2 
1 
o 

— 1 

— 2 
-3 



Sum 



i. 


("2i+c-!-m i 


£2i-C- 


m , i 


5 


— '00000 


0002 


+ '00000 


0204 


4 


— 'OOOOO 


0159 


+ ■00001 


2849 


3 


—•0000 1 


0382 


+ -00066 


4341 


2 


— •00056 


4777 


+ ■02214 


"77 


1 


—•02154 


4824 


+ ■02587 


20248 





— •26294 


86805 


— -08121 


00493 


— 1 


+ "04040 


239 8 3 


— •00205 


8170 


— 2 


+ •00196 


1854 


— -00005 


0845 


-3 


+ '00006 


349 2 


— 'OOOOO 


°753 


-4 


+ -00000 


0984 


— 'OOOOO 


0009 


-5 


+ 'OOOOO 


0013 






Sum 


— •24264 


0083 


— •03462 


9230 



Value of A. Coefficients of 



/2t+C-D 



+ •00000 
+ 'OOOOO 

+ -00006 

+ •00341 
+ •11582 
+ •23207 

— ■02089 
—•00037 

— 'OOOOO 

— •00000 

— "OOOOO 



0016 

1031 

557° 
6192 

9253 

94821 

28787 

5417 
9929 
0149 

0002 



^'" c 



— 'OOOOO 

— 'OOOOO 

— 'OOOII 

— •00414 

—•04792 

+ •09218 

+ •01113 

+ -00032 
+ -ooooo 
+ -ooooo 
+ -ooooo 



0032 

2037 
0038 

1602 

22751 
67678 
8493 

6072 

499S 
0066 



+ ■33011 3168 



+ ■05148 041 1 



Motion of the Moon. 



105 



The solution is 

tt^-^aee'S,^),; a+c+m + (c'»,')«<: 2 '^ m + HU 2i+c ' m + («W~° +,B ]» 

the first two terms forming oae set of equations of condition, and the other 
two another set. 

Values of 



i. 


("») 4 - 


(«V),. 


5 
4 

3 

2 
1 



— 1 

— 2 

-3 

-4 
-5 


— •00000 00016 

— •ooooo 01635 

— •00001 60440 

-■00143 5419 
—■09352 2778 

— •37910 7012 
-•00035 !5° 2 
+ •00000 14383 
+ •00000 00149 

+ '00000 OOOOI 


+ •00000 00023 
+ "ooooo 02402 

+ "00002 40337 

+ •00223 19829 
+ •16122 2282 

+ T45 I 5 115 
+ •00011 3163 

— •ooooo 09258 

— -ooooo 00095 

— "COOOO OOOOI 


Sum 


-•47443 1467 


+ •30874 192 



i. 


(«»'), 


(*'")r 


5 


+ •00000 OOOOI 


— 'ooooo 00003 


4 


+ '00000 OOIOI 


— 'ooooo 00346 


3 


+ '00000 10020 


— •ooooo 34589 


2 


+ '00009 30969 


-'00032 25311 


I 


+ •00721 48506 


-•02437 4803 





+ •12769 0229 


-•22224 55 2 


— I 


+ •03961 720 


—•00035 7897 


— 2 


+ •00004 6236 


+ •00000 62531 


-3 


— •ooooo 01813 


+ 'ooooo 00632 


—4 


— •ooooo 00020 


+ "ooooo 00006 


-5 






Sum 


+ •17466 244 


-•24729 793 



Royal Asteon. Soc, Vol. L1II. 



T06 



Mr. Ernest W. Brown, Theory of the 



The short period inequalities having small divisors are those with 
coefficients (e V ) , (*y)o ; («*)_!, (eV)i 5 («/)* («fy)oJ (o/)_i, (^)i- For 
the purpose of obtaining these with the required accuracy, the corresponding 
coefficients in A are carried one place further than the rest. The values of A 
have been computed by both (17), (17') of § 20. The slow approximations 
to the values of the coefficients with suffix zero were avoided by the method 
of § 29.* 

The values of the coefficients of the corresponding terms in the true 
longitude have been published in a note in the Monthly Notices, Vol. LV. p. 4. 

Section (iv). Characteristic e n . 

57. Here 21 + T has the two sets of values 2«'±2m and 21. The terms are 
similar in form to those of Section (ii). 



Values of A. Coefficients of 



i. 


£2t+2m_ 


C ai-Sm, 


C M - 


5 






+ •00000 


003 


— "OOOOO 


OOI 


4 


+ -ooooo 


003 


+ 'OOOOO 


166 


— 'OOOOO 


052 


3 


+ -ooooo 


236 


+ -00008 


791 


— -O0OO2 


893 


2 


+ •00010 


062 


+ -00308 


°35 


— '00113 


096 


1 


+ '00208 


790 


+ •01194 


981 


— '01521 


048 





+ •01086 


3018 


+ -00684 


6387 


-•03584 


293 


— 1 


— •08426 


889 


— •00006 


164 


+ •02561 


810 


— 2 


+ •00013 


885 


+ -ooooo 


798 


— -00005 


617 


-3 


+ 'OOOOO 


733 


+ 'OOOOO 


019 


— -ooooo 


228 


-4 


+ 'OOOOO 


013 






— 'OOOOO 


003 


-5 














Sum ... 


— •07106 


865 


+ •02191 


268 


— ■02665 


421 



* The results for the coefficients in A are not theoretically accurate in the last places of 
decimals given. The small divisors occurring in the coefficients mentioned and the other 
divisors are of such a size that the results for the coefficients («;) a &c, are, however, 
theoretically accurate to the last place given in each case. 



w 



Motion of the Moon. 
The solution is 

which gives two sets of equations of condition as in Section (ii). 

Values of 



107 



i. 


M,- 


(V% 


(rm%. 


5 
4 
3 

2 
I 



— I 

— 2 

-3 

-4 

-s 


+ 'ooooo 00004 
+ 'ooooo 0035 
+ •00000 2205 
— •00003 2 77 
-•05446 177 
-•10598 405 
+ •00001 0669 
+ •00000 0238 
+ •00000 00026 


+ - ooooo 00003 
+ •00000 00290 
+ •00000 2938 
+ •00027 6 5 86 
+ •02192 232 
+ ■07221 455 
+ •00007 2 ^7 
+ •00000 0016 
+ -ooooo 0003 


— 'OOOOO OOOOI 

— 'ooooo 00083 

— 'ooooo 08202 

— •00007 6201 

-•00585 014 

-•01024 957 
+ •02515 958 
—•ooooo 1878 

— •ooooo 0060 

— '00000 00005 


Sum 


— •16046 544 


+ ■09448 911 


+ -00898 090 



The long-period inequality with a small divisor is that having the 
coefficients (rf) , (y' 2 )o- To obtain it with sufficient accuracy, the homo- 
geneous equation (7) or (42) of Chap. I. was calculated for 2z'+T=2tn, and 
combined with one of the equations of condition of principal importance in 
finding these coefficients by the above method ; the corresponding terms in 
A are carried one place further. The slowness of the approximations was 
avoided as before. 



Section (v). Characteristic k 2 . 

58. Here ii + r has the two sets of values, 2r±2gand 21. The forms 
are similar to those of Sections (ii), (iv). 



io8 



Mr. Ernest W. Brown, Theory of the 

Values of A. Coefficients of 



i. 


C 2i+2 *. 


£3t-2g. 




C 2i . 


5 


+ '00000 


0001 


4- "OOOOO 


0005 


+ -ooooo 


0005 


4 


+ "00000 


0098 


+ -ooooo 


0373 


+ -ooooo 


0420 


3 


+ -ooooo 


8104 


4- 'OOOOI 


9128 


+ -00003 


0535 


2 


+ - ooo6i 


3392 


4- -00019 


5705 


+ -00179 


3676 4 


I 


+ '03948 


0728 


- -09576 


3403 


+ -05619 


8303 


O 


+ 175467 


4677 


+ 1-75605 


S74S 


-S'S 1 ^ 


8315 


— I 


- -11368 


2769 


+ -02156 


4305 


+ -09211 


3861 


— 2 


+ -00119 


0891 


+ '00022 


7392 


+ '00118 


5620 


-3 


+ -ooooi 


6227 


+ 'OOOOO 


2257 


+ -ooooi 


2707 


-4 


4- -ooooo 


0178 


+ -ooooo 


0022 


+ -ooooo 


0126 


-5 


+ -ooooo 


0062 






+ -ooooo 


0001 


Sum 


+ 1 '68230 


*529 


+ 1-68230 


1529 


—3-36460 


3061 



The solution is 



t^- 1 =ak 2 2,((^) i C 2i+2g +(A' 2 )^ 2i - 3g + (M')^ 2i ]. 
Values of 



i. 


<**>«• 


(# 2 V 


(»'),• 


5 






+ 


•ooooo 


ooooo 8 


— 'OOOOO 


ooooo 1 


4 


+ -ooooo 


OOOOO 2 


+ 


•ooooo 


00096 


— -ooooo 


00014 9 


3 


+ -ooooo 


00009 


+ 


■ooooo 


I33 01 


— -ooooo 


01983 


2 


+ -ooooo 


01113 


+ 


•00020 


4729 


— -00002 


78210 


1 


+ -oooo I 


40450 


+ 


•04329 


3868 


— -00434 


42967 





+ •00165 


67611 


+ 


•98752 


5842 


— 1-00079 


9130 


— 1 


— -09302 


7702 


+ 


■00150 


88256 


+ -08149 


6924 


—2 


+ •00081 


6246 


+ 


•ooooo 


58653 


+ -00009 


25048 


-3 


+ -ooooo 


14413 


+ 


•ooooo 


00300 


+ -ooooo 


03390 


—4 


+ -ooooo 


00059 


+ 


•ooooo 


00001 8 


+ -ooooo 


00016 6 


-5 


+ -ooooo 


ooooo 4 








+ -ooooo 


OOOOO I 


\ Sum . . . 


-•09053 


9090 


+ 


[■03254 


0500 


- -92358 


1678 



Motion of the Moon. 



109 



These coefficients were given to seven places of decimals by Mr. P. H. 
Cowell on pp. 119, 117 of his paper referred to in § 3 1 above, being obtained 
by means of the homogeneous equations. With his values as a first 
approximation, I recalculated and extended them to nine places by the above 
method. Small errors in (k*)_ u {l J% \ were found. These two coefficients 
are those of a long-period term with a small divisor. The former was 
obtained by means of equation (44) of Chap. I., the latter being then 
found from one of the two equations of condition furnished by the general 
method used for the rest of the coefficients. The coefficients {kK) t in 
Mr. Cowell's paper are such that (k¥) =o ; to compare them with those given 
here, a transformation like that noted at the end of Sect, (ii) of this chapter 
must be made. The value of Sa in that formula is the (M 7 ) of the table in 
this section. 

Section (vi). Characteristic ea. 

59. Here 2i + T=2i 1 ±c where 2% is an odd positive or negative integer. 
It is not necessary to insert the suffix of i x in the tables. 

Value of A. Coefficients of 



2i. 


f2i + C 




p-e. 


9 


+ '00000 


002 


+ '00000 


010 


7 


+ 'ooooc 


053 


— '00000 


676 


5 


— •00003 


223 


-•00135 


769 


3 


— •00682 


665 


— 'IOOI0 


7433 


1 


-•51730 


75492 


+•13499 


37257 


— 1 


+ •14791 


32045 


— -I00I2 


41023 


-3 


— •00226 


4428 


—•0065s 


106 


-5 


— '00141 


822 


— '00004 


061 


-7 


— •00000 


864 


— -ooooo 


022 


-9 


— '00000 


004 






Sum 


-■37994 


400 


-•07319 


40s 



The solution is 



M^-^aeaS^aXP'^+Ce'a)^'-'], 2% odd. 



no 



Mr. Ernest W. Browk, Theory of the 

Values of 



2i. 


(«)«• 


<«'»),. 


9 


+ '00000 


00001 


+ 'OOOOO 


00007 


7 


+ "OOOOO 


O0I2I 


— 'OOOOO 


0210 


5 


+ "ooooo 


0067 


— - oooo6 


845 1 


3 


— •00018 


3543 


-•01423 


8397 


i 


-•0467s 


5060 


-•13023 


797 


— i 


+ •19695 


832 


— •01226 


5219 


-3 


+•01753 


2892 


—•00059 


1716 


-s 


— •00014 


55" 


— "OOOOO 


1318 


-7 


— -ooooo 


0318 


— 'OOOOO 


000 10 


-9 


— 'OOOOO 


00008 






Sum 


+ •16740 


685 


-•15740 


328 



The long-period inequality is that having (ea).^ (^0.)$ as coefficients. 
The homogeneous equation (42) of Chap. I. with ii+r= — i+c was used 
with one of the two ordinary equations of condition giving these two co- 
efficients, the other being, as usual, used as a control. The slow progress of 
the approximations was avoided as before. 

Section (vii). Characteristic efa. 
60. Here 2i + T=2i 1 ±m. The suffix of i x will be omitted. 

Value of A. Coefficients of 



2i 


c- i+m - 


fti— m # 


9 






+ 'OOOOO 


003 


7 


+ -00000 


012 


— 'OOOOO 


138 


5 


+ -00004 


540 


-•OOO33 


248 


3 


+ '00405 


8824 


— •O2654 


623 


1 


+ ■06010 


42164 


-•05865 


88010 


— 1 


— -04094 


6441 


+ •01713 


45698 


-3 


-•06584 


676 


+ ■01489 


4425 


-5 


— •00036 


75 2 


+ -0O006 


690 


-7 


— 'OOOOO 


211 


+ 'OOOOO 


036 


-9 










Sum 


-•04295 


427 


-•05344 


261 



Motion of the Moon. 



in 



The solution is 

u e , ^ 1 =ae'aS,-[(i?a) 1 .^ +m + (Va) i (; 2{ - m 3, ' 2% odd. 
Values of 



2i. 


(i a ) r 


(V«),. 


9 


+ "0000O 


000 10 


+ "OOOOO 


00016 


7 


+ '00000 


0123 


+ 'OOOOO 


0126 


5 


+ "00001 


8262 


+ 'OOOOO 


8022 


3 


+ •00286 


0184 


+ -00008 


9662 


i 


+ •51611 


841 


— -02661 


542 


— i 


+ •03082 


496 


— 1-51100 


393 


-3 


— '01661 


267 


+ -00301 


4686 


-5 


— •0000 1 


8246 


+ -ooooo 


443i 


-7 


— "00000 


0037 


+ -ooooo 


0015 


-9 


+ 'OOOOO 


0000 1 


+ -ooooo 


0000 1 


Sum 


+•53319 


099 


-i'5345° 


241 



The shorfc-period inequalities are those having the coefficients (i?a)j, 
(i/ a ) _ 4 ; (^a) _ 4 , (ij'a)j ; the former pair, owing to the near coincidence of 
1 + m with c , having a very small divisor. The corresponding values of the 
coefficients in A are carried to more places and the slow progress of the 
approximations to the first pair was avoided as before. 

Section (viii). Characteristic a 2 . 
61. Here t=o. 

Value of A. Coefficients of 



i. 


C n - 


4 


— -ooooo 01 


3 


— -OOOOI 02 


2 


-•00025 53 


1 


+ •05977 63 





-•03355 85 


— 1 


+ •00220 05 


— 2 


— •01280 45 


-3 


— '00006 20 


—4 j 


— -ooooo 04 


Sum ... ! 


+ ■01528 58 



ii2 Mr. Ernest W. Brown, Theory of the 

The solution is 

Values of 



i. 


(«% 


4 


+ •00000 0007 


3 


+ -ooooo 097 


2 


-f -00009 385 


I 


+ •00722 77 


O 


— •00960 28 


— I 


— •00720 42 


— 2 


—•00142 707 


-3 


— •00000 223 


-4 


— - ooooo 0008 


Sum 


— -01091 38 



These coefficients were obtained to six places of decimals on p. 157 of 
my paper referred to in § 50 above by using the homogeneous equations. 
They have been recalculated by the method of this chapter and extended to 
seven places. In comparing the earlier results with those given here, the 
transformation noted at the end of Sect, (ii) of this chapter must be made. 
The errors in the earlier results were very small. 

62. Having finished the terms of the second order in u, we now come to 
those in z which are distinguished by having the first power of k in all their 
characteristics. For these terms the equations (s)-(8) of Sect, (i) of this 
chapter are used. The calculations cause very little trouble and are not 
long. In the short-period inequalities with small divisors, the progress of 
the approximations is not very slow. Long-period inequalities do not pro- 
duce small divisors. 

Section (ix). Characteristic ke, 

63. Here 2«" + t takes the two sets of values ±(2& + g + c)and ±(2i + g-c), 
each set giving an independent set of equations of condition for the co- 
efficients. 



Motion of the Moon. 

Values of A. Coefficients of 



i. 




ftt+g + c 






f ,,- + g_e 




5 


+ 


•ooooo 


0001 


+ 


•ooooo 


OOIO 


4 


+ 


•ooooo 


0054 


+ 


•ooooo 


0817 


3 


+ 


'OOOOO 


4225 


+ 


•00006 


2854 


2 


+ 


•00032 


0242 


+ 


•00398 


8176 


I 


+ 


•0204s 


4910 


+ 


•16874 


9622 





+ 


•87305 


5636 


+ 


•86734 


1740 


— I 


+ 


•13526 


5495 


_ 


■01316 


9637 


— 2 


— 


•00250 


6140 


— 


•00046 


2414 


-3 




•00008 


9740 


- 


•ooooo 


7977 


-4 


- 


•ooooo 


1579 


- 


•ooooo 


0106 


-5 


— 


•ooooo 


0019 


— 


'ooooo 


0001 


Sum 


+ i 


•02650 


3085 


+ 1 


■02650 


3084 : 



Coef. of £ 2i -s-°=_coef. of £-*+*+°, 
Coef. of f"-s+°=_ coef. of t !i4 n 

The solution is 

« ket =akeS,[(eA) j ^ + « + « + ( e '*')(^~ g " c +^).ff ! ' + *" c + (^')^" g * ] 
=akoS ( [(£&) ( (£* + » +e - t 2i ~ g - c ) + (e^M****-* - r 2 ^ g+0 )]. 
Values of 



i. 


(J<)i. 




(« 


74-. 




5 






_L 


ooooo 


0000 1 


I 


i 4 


+ 


ooooo 00006 





+ 


ooooo 


00149 


3 i 


*> 
1 


+ 


ooooo 00740 


3 


+ 


ooooo 


20814 


9 


2 


+ 


00001 00784. 


2 


+ 


00030 


77463 




1 


+ 


00146 61399 




+ 


05543 


3972 







+ 


2 5°9 I 3591 




_ 


73687 


7762 




— 1 


— 


1 1999 3099 




- 


008 1 1 


4895 




\ —2 


— 


00x79 7078 




- 


00004 


46815 




! " 3 


- 


ooooo 93584 




- 


ooooo 


03014 


9 ! 


-4 


- 


ooooo 00626 


5 


— 


ooooo 


00021 


4 


— c 

.J 

Sum 


- 


ooooo 00004 


I 


— 


ooooo 


ooooo 


I ' 


+ 


13059 0286 


— ■68929 


3827 





There is no short-period inequality with a small divisor. 
Royal Asteoh. Soc. Vol. LIII. 






Mr. Ernest W. Beown, Theory of the 



Section (x). Characteristic ke'. 
64. Here 22' + t has the two sets of values ±(2i + g + m) and 
giviDg rise to two independent sets of equations of condition. 

Values of A. Coefficients of 



K 2i + g—m] 



i. 


/■SiH-g^-m^ 


f2i + g- 


m j 


5 




+ 'OOOOO 


0002 


4 


— -ooooo 0034 


+ 'OOOOO 


0222 


3 


— •00000 2530 


+ •00001 


7 26 3 


2 


— •00016 8839 


+ •00114 


5257 


1 


-•00794 5172 


+ •05318 


2849 





— •00150 67489 


— •00386 


17596 


— 1 


+ •05329 66967 


-•00788 


15823 


— 2 


— •00090 3017 


+ •00013 


7147 


-3 


— 'OO002 6532 


+ 'OOOOO 


39 2 3 


-4 


— -ooooo 0433 


+ 'OOOOO 


0061 


-5 


— -ooooo 0006 


+ 'OOOOO 


OOOI 


Sum 


+•04274 3385 


+ •04274 


3383 



Coef. of ^"8~ m = -coef, of £-«+«+» 
Coef, of £»-«+»>=: -coef. of {~*+g~-<*. 
The solution is 



-£-2*-S-'») + (^'£) ( (£!>+g--m_£-a(--g+m)J_ 

Values of 



i. 


(#)»• 


(V'k)i. 


5 






+ "OOOOO OOOOO 


2 


4 


— -ooooo 00004 


s 


+ *ooooo 00030 


2 


3 


— •ooooo 00549 


5 


+ ooooo 03925 


4 


2 


— •ooooo 71590 




+ •00005 16918 




1 


— •OOO92 26l 12 




+ •00680 83683 







— -01600 9252 




+ •01924 4655 




— 1 


— "II002 672I 




+ '04091 0818 




— 2 


— -00033 46147 




+ -00008 41 1 28 




! -3 


— ■ooooo 20106 




+ -ooooo 04271 




[ —4 


— -ooooo 00139 


3 


+ •00000 00026 


7 


-5 


— 'OOOOO OOOO I 





+ *ooooo ooooo 


I 


Sum 


— •12730 2438 




+ "06710 0471 





Motion of the Moon. i 1 5 

The short-period inequalities having small divisors are those with co- 
efficients (rjk) 0) (y]k)_i, (t]'k) , (V^)-i5 for these the corresponding values of 
A, are carried to ten places of decimals. 

Section (xi). Characteristic ka. 
65. Here 2/ + T =±(2? 1 + g). The suffix of/, will be omitted. 

Value of A, Coefficients of 



2i. 


C' 2(+g - 


- _ __. 

9 
7 
5 
3 
1 1 

! _1 

i ~ S 
i -7 

! - 9 


+ 'ooooo 0008 
+ ■00000 0534 
+ •00002 6082 
— '00008 7162 

— •19302 7070 

--18559 984? 
+ -00734 5267 

+ '00002 8303 

— 'ooooo 0473 

— *ooooo 00 1 3 


Sum 


■-•57131 6371 



The solution is 



Coef. of £*-*=- coef. of t" li ^.. 



£^i=aka2 ; (ia) i (^'^-f-- i - 6 ;. zi otlll. 



ix6 Mr. Ernest TV, Brown, Theory of the Motion of the Norm, 

Values of 



j 

2i. 


(&»),. 


I 


9 


+ 'OOOOO ooooo 


^ '• 


7 


+ *ooooo 00072 


1 


5 


4- - ooooo 05027 




3 


-■00005 33463 




i 


— •06026 0507 




— i 


4- '15913 4186 




-3 


+ •00375 4653 




! -s 


• 4- -ooooo 67585 




: -7 


4- -ooooo 00082 




-9 


— 'OOOOO ooooo 


9 


Sum 


4--I0258 2262 

! 



There are no short-period Inequalities with small divisors, 

(To be continued,) 



J ppendix.— The terms of the third order (Chap, V.) are in process of calculation. The 

following results, properly belonging to Chap. V., have already been obtained and may ht 

recorded here. They have been used as the basis of a paper in the Monthly Kotie™, "Or, the 

Mean Motions of the Lunar Perigee and Node" (1897 March). 

Let 

e = e„ + e 2 c e , + e'"e i; „ + k 2 c k3 , 

g = g. + e 2 §V- + «'%,/= + k2 g k - 

Then 

c e _=+ -00268 575, g e ,= +-00318 579, 

%.,= --03465 53= g e --= +-00564 65, 

e^= +'05384 91, g t ,= --00806 633. 

The value of g k » is quoted from Mr. Co-well's paper referred to in § 31 above. 

Ilaverford College, Pa, U.S.A. : 
1897 May 12. 



Theory of the Motion of the Moon; containing a New Calculation of the Expres- 
sions for the Coordinates of the Moon in Terms of the Time. By Ernest 
W. Brown, M.A., Sc.D., F.R.S. 

[Received 1899 February 3.] 



PART II. CHAPTER V. 

In the following pages I continue the Memoir the first part of which was 
published under the same title in the Memoirs of the Royal Astronomical 
Society in 1897. The general theory was given in Chap, I. as completely 
as I could then foresee would be necessary for the whole work. In 
Chaps. II., III., IY. the numerical results up to and inclusive of the terms 
of the second order were given. 

As the work progressed modifications tending to simplify or abbreviate 
the calculations naturally occurred. These, however, were fewer than might 
have been expected. The most important of them is given in Section (ii) 
below, consisting of a new method for finding the values of the final 
coefficients, after those of the quantities denoted in Chap. I. by A have 
been obtained. Previously this process consisted in solving, by continued 
approximation, for each characteristic and argument, a set of linear equations 
which were generally about 20 in number, with 20 unknowns. This process 
I had not succeeded in arranging conveniently for the computer, and as, in 
the terms of the third order, it involved about one-third of the whole work, 
some change was desirable. The investigation which led up to the new 



164 Mr. Erxest AV. Browx, Theory of the 

method for tliis purpose was made from a different point of view some three 
years ago ; its usefulness became apparent directly the arrangement of the 
work for the computer was under consideration. Moreover, the numerical 
errors made in solving the linear equations were quite numerous ; under the 
method of Section (ii) they have been no more frequent than in other parts 
of the work. 

The numerical results given below are the values for all the terms of the 
third order, with certain subsidiary results which the above-mentioned modifi- 
cations require. The degree of accuracy which the theory up to this point 
attains may be best appreciated by a statement of the maximum number of 
coefficients of the fourth and higher orders, which may be as great as 1" of 
arc in longitude. Of the fourth order, one of 4" and three of 2", which 
contain only e, k in their characteristics, and twenty of 1" ; of the fifth order 
there are two of 1", which involve both e, k. Moreover, the principal parts 
of nearly all. of these are the purely elliptic terms. 

I have received very great help in performing the calculations from 
Mr. Ira. I. Sterner, A.B., of Haverford College, who has, since 1897 Septem- 
ber, been my only assistant. That so much has been achieved in the time 
we were able to give is largely due to his accuracy and capability. Much of 
the work done by him would scarcely have been attempted by an ordinary 
computer without very extended instructions, while his knowledge . of 
arithmetical processes has not only been a great saving of time and labour, 
but has made the chief part of my task — that of testing and correcting his 
work — a comparatively light one.* 

The following is the table of contents of Chap. V. : — 

Section (i). A brief outline of the application of the general method to 
the terms of the third order in the calculation of the series A. 

Section (ii). JS T ew method for solving the linear equations when the 
series A have been obtained. Numerical values of certain quantities required 
in this method. 

Section (iii). Modification of this method, in order to avoid, as far as 
possible, the loss of accuracy arising with long-period terms. 

Section (iv). The method of calculating the new parts of the motions of 

* A portion of the expense of making the computations necessary to obtain the results 
given below has been met by a grant from the Government Grant Fund of the Royal Society. 



Motion of the Moon, ^5 

the perigee and node, and the coefficients arising therewith. Numerical 
values of certain quantities required. 

Section (v). The final numerical results for the series A, and for the 
coefficients of all terms of the third order in v. :.* 



Section (i). Formula' and 7 est*. 

66. About two-thirds of the whole labour of obtaining the coefficients of 
the third order consists in the calculation of A. The products in the third 
line of equation (17) and of the second line of (18) in Chap. I., are formed 
by putting 

and choosing out the parts of the third order. The products in the fourth 
and fifth lines of (17) and the third line of (18) are obtained with 

Mfi = W<l, s p— s l> % — ^1 

Here 

ll^lle + U^ + Ua, »i=%, 

U.f= U e , + U ee , + U e ,~. + U w + W e . -f U lU + It,,, 

the expression for s 2 being similar to that for n 2 . 

The parts arising from 8 1} namely.-— 1 £-\ - $° Q \ are treated in like 

os dz 

manner. 

The parts arising from the first terms in the right-hand members of the 
equations, are treated in Section (iv) below ; they only appear when the 
terms of arguments 2i±c, 21 ±g are under consideration. 

The general method of procedure has been as follows : — The computer 
having performed the calculations allotted to him, I go over them all and 
test them by all the means available. Each multiplication of series is tested 
by sums as explained in Section (viii). Chap. I. The final values of A are 
tested in the same manner, but in larger groups, so as to make certain that no 
series of terms has been omitted ; for this, the values of A for each character- 

* It is intended to give the subsidiary results in a final chapter, or chapters, when tlio 
whole theory has been completed. 

Royal Astros. Soc, Vol. LIII. a a 



ra 



1 66 Mr. Ernest W. Brown, Theory of the 

istic, with £= i , are all added together and the sums compared with the 
corresponding sums obtained as directly as possible from the algebraical 
formulas, which become quite simple when £= i . It is true that this method 
will not test for all kinds of errors, e.g. the accidental interchange of u and s, 
but reliance has been placed less on test equations than on the care which has 
been taken to avoid errors. The calculations made after the method of 
Section (ii) were treated in the same way as far as possible ; a further and 
very searching test was obtained by forming Dh/ M Dn A with ?=i, for each 
value of r and substituting the results in equation (24) of Chap. I. In the 
cases where calculations were not turned over to the computer, they were 
gone over again after an interval of time and, when possible, tested. 
Increased accuracy in making the calculations was always found whenever 
blocks of the same nature were performed together. 



Section (ii). Method of solving the Linear Differential Equations. 

67. The equation (24) of Chap. I. to be solved is 

(D + myu x + Mu x +A r s x c, i -a.XAi:=A' ... . (1) 



Let 






Then u/, s/, j=i, 2, 3, form three particular integrals of (1) when .4=0. 
In a paper to be published elsewhere, I have shown that the fourth particular 
integral can be given the forms 

t l =u/=u 3 ' J D- 1 ( --t-,- 1 

V u 3 s/ / 

=Iu.' r S -L U2 -_ u >' s A + 73-i f Ca s/uZ-u.V / m , D\i a ' _DaA\ m l 
2 ° L <s/ i U 3 'S 3 ' U 3 'S 3 ' V U/ S//IJ 



«=s 4 '=u/, 



where C 12 is a constant given by 

C K =s 2 'i)u/-u 1 'Z)s/+u 2 'i)s 1 '-s/i)ii/-2m(s 1 'u./-ti l 's 2 ') 
= 25(21 + 1 +m + c ) 2 i + 23(2* - 1 - m + c )f_i' 2 , 

the bar over u/ denoting that %~ x has been put for £ that is -< for t in the 
expression for u/. 



Motion of the Moon. iQy 

It is further shown that 



C 



-=u 3 'i)- 1 q + u^=U3' t (M-M')(<-gq + u i ; ) 



where q is the constant term uncfer the sign D l in the expansion of u 4 70 12 
in powers of ^ and u 4 £ is a series of the same form as u 3 . 

Finally, it is shown that the solution of (i) when A is not zero can be 
put into the form 

Mi= tJ7 2 [ u i' Z) ~ I ( s *'^'+ Us'i'O-Uj'D-'foM' + u,'!')] + u 4 ;Z>~- 1 (s 3 M' + u 3 'I') 

-u.'n-ms^A' + n.UO-qD-^sW+uM')} (2) 

in which the bar over A has the same meaning as before. 

This is the form required ; it will be noticed at once that most of th 
operations will consist of multiplications of series, and will therefore be h 
line with the work which was necessary for the calculation of A. 

68. It is advisable to make a few changes in order that the formula 
may be more compact. We first observe that for all purposes except the 
parts due to S3 , we require, not u„ but u x /u ; the parts due to £3 have the 
factor m 2 , and are therefore small, and the multiplication of u x /u by u is 
easier than that of u % by i/u owing to the fact that u v has one coefficient 
unity. As there is no increase of trouble involved in finding uju m we shall 
do so. 

Write therefore — 

Uj , u 2 8l s,. (3) 

*o *o £q *q 

U 3 =U 3 'f- 1 , 85 = 83'^ (4) 

tt _af V^ o<> T t _ae 'u 2 '4' c ' > TT _au 3 ' TT au 4 ; , . 

These ten quantities are then all series of the form 2 jt? ( £ 2i with numerical 
coefficients, and the constant coefficients in u 2 , s 3 , U 3 , are all unity. 

Also, D- 1 (X£°) may be written £° (D + c)-*X; the operation D^ % 
consisting of a division by t, the operation (D + c)~ l £ T will consist of a 
division by t + c. 

The formula (2) may now be written 

X ^=U l (2)-c )- 1 (s 2 ,i + ii 2 Z)-U 2 (D + c )-\s l A+u l A) + -U i I)- 1 (s 3 A+u 3 A) 

-U 3 /)- 1 {(M + u l I)- q _Z)^ ( S3^+u 3 Z)!- ( 6 ) 



1 68 Mr. Ernest W. Brown, Theory of the 

Next, put 



Then since 



W>= Si A + u 4 J, T^D^'qY.-W,) 



S,— U,, 11;— S 2 , u 3 — -S 3 , U 4 = s 4l D——D, 



(7) 



and therefore 

-(D + Co^Xs^l + UjJj^Q,, UjJ"=-M. u ! .r=s7I7 V^V,, TY=W„ T»=-T„ 

the equation (6) takes the final form 

1 ^•=TJ 1 Q x + UA + TJ 3 T i + U 4 V !l (8) 



J 



69. The calculations are arranged as follows : — The series u 2 , s 2) s 3; s 4 , 
Ux, L T 3 , U 3 , U 4 are obtained once for all by multiplication or " special values," 
and the logarithms of the coefficients written out on slips, as explained in 
Chap. I., Section (viii). The slips containing A having been made out for 
each value of \ the multiplications 

s. 2 A, u.,i, s s A, s 4 i (9) 

are performed by the computer, and thence the values of Q A , Y A , T A are, by 
a few easy processes, obtained. The slips containing these latter are then 
made out, those for Q A serving for Q A . Finally the multiplications 

U,Q« U 2 Q-„ U 3 T„ u 4 V> ... ... (10) 

are performed by the computer, and thence, by addition, the value of uju is 
obtained. 

The series A are of the form 2wp,£ 2i+T + £ (j p,-£ w ~ 7 . When t is not zero, the 
proc is for each such double set involves sixteen multiplications of series 
of tl : form (Sp^ 2 ')^^ 2 '). When t=o, half the number suffices. When 
r = c,; a new part of the motion of the perigee is under consideration, and 
though the same process may be used, it is not convenient (see Section (iv) 
below). 

When t=o and 21 is even, it would appear that T A gives rise to terms 
with the time as a factor. It is shown in the paper referred to at the 
beginning of this section that such terms can always be made to disappear. 
However, none such occur amongst the terms of the third order. In any 



Motion of the Moon. 



169 



case the method of this section will not be used for them, as the approxima- 
tions in the ordinary method are rapid, owing to the absence of any small 
divisors. 

The same method may be applied, using well-known formulae, to the 
calculation of z k . but it appeared easier, at any rate in the terms of the third 
order, to use the method given in Chap, I. 

The chief objection to, the method consists in the fact that the small 
coefficients which accompany large values of i appear as differences between 
comparatively large 'numbers. This fact does not impair the required 
accuracy of the results, but the multiplications are much longer than they 
otherwise would be. Thus the coefficient of ^ 6 ~ 3 '" in u e .-./e 3 t/ appears as 
follows : — ■ 

From UjQ, — '00006 418 

,1 VzQn +'OOOo6 496 

„ U 3 Tj + 'ooooo 316 

» "C 4 V, — -ooooo 39 2 

Sum + 'ooooo 002 

On the other hand, as the presence of this difficulty can be shown to be 
peculiar to the method, it furnishes a means of detecting a certain class of 
error. 

70. The following are the numerical results required for the method of 
this Section : — 









Values of the coefficients of £ 2 ' in 








i 


s,,. 


■a,,. 


%■ 


s v 


4 






— 'OOOOO 


002 










3 


— '00000 


017 


— 'OOOOO 


279 


+ -ooooo 


0012 


— 'ooooo 


003 i 


2 


+ '00006 


960 


— -00041 


211 


+ -ooooo 


0491 


— 'OOOOO 


147 1 


1 


+ ■19895 


557 


— -07424 


980 


— -00869 


5747 


+ •03592 


927 i 





-•33619 


118 


+ 1 




— I 




-•93146 


358 1 


— 1 


— '00196 


357 


+ -00016 


930 


- -00454 


7122 


— '00648 


I48 ! 


— 2 


— 'OOOOI 


349 


— -ooooo 


090 


— -00002 


9393 


— -00004 


410 j 


-3 


— '00000 


010 


— -ooooo 


001 


— -ooooo 


0210 


— 'OOOOO 


032 ! 


—4 










— -ooooo 


0002 






Sum 


- -13914 


334 


+ "9255° 


367 


— 1-01327 


1971 


—-•90206 


171 • 



170 



Mi\ Ernest W. Brown, Theory of the 



Values of the Coefficients of £ z * in 



i 


U,. 




u 2 . 




U,. 




u. 


4 






— 'OOOOO 


003 










3 


— "OOOOO 


015 


— 'OOOOO 


417 


+ 'OOOOO 


013 


— '00000 


022 


2 


— •00002 


034 


— '00066 


197 


+ - OO0OI 


892 


— "00003 


094 


1 


—•00317 


658 


— '16544 


214 


+ -00303 


148 


— •00506 


996 





-•73476 


98s 


+ 2-18209 


t-5 


+ 1 




-•93154 


984 


— 1 


+ ■42803 


606 


+ -01934 


433 


+ -01739 


126 


+ •02782 


838 


— 2 


+•00387 


413 


+ 'ooo 1 6 


59° 


+ "00015 


057 


+ -00024 


067 


-3 


+ '00003 


326 


+ 'OOOOO 


142 


+ 'OOOOO 


129 


+ -ooooo 


206 


-4 


+ 'OOOOO 


029 


+ 'OOOOO 


001 


+ 'OOOOO 


001 


+ "OOOOO 


002 


Sum 


— •30602 


318 


+ 2-03549 


49 


+ 1*02059 


366 


-•90857 


983 



q= + i-5i4i5 29. 

71. In the terms of the third order the above process has been actually 
used. In those of higher orders a slight abbreviation of the work will be 
made by the following change. 

Put 



A _ 3 M *0-> A 

Po 



(10). 



and, instead of u 2 , s 2 , s 3 , s 4 use each of these series multiplied by Zw<£~ l /4pi. 
The latter four new series can be obtained once for all. 

The advantage gained from this change will be seen by looking at 
equation (17) of Chap. I. The terms A consist of two parts. The first is 
the part due to the first line of the second member of the equation ; the 
terms in this line are small, being due to Sc and £3, and their multiplication 
by APo/ZW-g- 1 to get the corresponding part of A 1 will be short. The 
second is the part due to the succeeding lines of the equation ; these always 
form the principal part of A, and they all have the above-mentioned factor. 

Section (iii). The Terms of Long Period. 

72. Small divisors arise when the period of any term approximates to 
that of the principal elliptic term or when it is long. In the former case, the 
small divisor arises in the first and second terras of (8), being due to the 



Motion of the Moon. 171 

operators (D±c )~ 1 , In the latter case, as was pointed out (Chap. I., § 28 (c)), 
the square of the divisor occurs, but this difficulty may be avoided by the 
use of the homogeneous equation (21). I shall now show how the latter 
equation may be adapted when the methods of the last section are used. 

The long-period small divisor arises only in V, T, owing to the operator 
D~ x . But (omitting the suffix X for brevity) 

T=Z>-'(qV-W), Y-=D~i(As, + Au 3 ), 

so that the square of the small divisor arises in the part 

D 1 (qV)=qD--(As 3 + Au s ) 

of T. This is therefore the term to be considered. 
We have 

BY=As 3 + Iu 3 =l(AiDs + Ai;- l Du ) 

=l{(A>:-hi, ti -A'Cs^ + Ai:--\Du ii -u a )-\-A2(Ds (l + s )) ... (11) 

But by equation (21), Chap. I., 

u D\ + u x D\—s D-u x — s x D :i u t) ~2mD (s u x + u 6 s x ) + ^vo?(u a u x — s u s x ) = a 2 A. A ', 

where A' denotes all the terms of characteristic a except those due to u K , s M 
in the expansion of 

D(sI)u-uDs + 2mus) + 3 m%s i -u i ) + s^ -J*® 1 txz) 

2 as On x ' 

This equation may be written — 

or, using equation (24) of Chap. I., 

u A£ - 1 — s a Ai=an\'. 
Whence, restoring the suffix a, 

Y x =D-K\ x ' + I D-^{A^\Du a -i k ) + A-C(Ds + s )} (13) 

(X 

where a 2 X I)"" 1 A/ will denote the known terms of characteristic X, in 

sDu-uBs+2raus + .D' l \ 3 m%s i -u i ) + s^-ui^) (14) 

1 2 ' Os Cu ) ' 



rm 



172 Mr. Erxest ~W, Brown, Theory of the 

The portion of (14) under the operator D^ 1 contains the factor m 2 at 
least ; hence the effect of the small divisor (which is never of an order higher 
than m 2 ) is neutralised in the first term of V. The same thing occurs with 
the remainder of the expression for V. owing to the fact that J)u —u and 
D$ +s contain the factor m 3 at least. Hence V can he found to the same 
degree of accuracy as A, and the loss of accuracy in u k is limited to that due 
to the first power of the small divisor. 

In using this we first find ^1 as usual, and with it calculate the coefficient 
of the particular power of ~ in the second part of (13). The terms contain- 
ing this power of £ for the given value of ?. are then chosen out of the 
expression (14), and thence V is obtained to the recpiired accuracy for this 
power of %. The method of Section (ii) serves for all the other powers of £. 



Section (iv). The New Part* of the Motions of the Perigee and Node. 

73. The method of calculating c A/e is explained in Chap. I., § 28 (b) ; 
when this quantity has been obtained A is completely known. Putting X 
= X ; (§25) and omitting one of the equations for i=o, we can solve the 
linear equations by continued approximation. The omitted equation serves 
as a test. Either this or the method of Section (ii) somewhat modified, can be 
used, but it is more convenient to proceed in the following manner. 

In § 28 (b), we have put A = B + c Ah b. Hence write the solution in 
the form 

v^v+XoC/'x+ov.co,, J ' '" "* 1IS) 

except for i=o. 

Here ^ jX/, are obtained by solving equation (26) of Chap. I. with 
A i =B i ,A[=B',-K Q =o=^, 

The terms {f\ 7i , (f\ X are obtained by solving the same equation with 
A i =o=A i f , x =A ' (§25), in terms of A . (If, instead of x =?. / , we had put 
^o—^o'— 1 ) we should have found s,., s/.) 

The terms (c),-, (c')i are obtained by solving with Ai—b i} J./ = 6/, 
A =O=A . 

In all three cases the equations for i=o are omitted. 



Motion of the Moon. 



^73 



We now substitute the values of \ h x' ft thus obtained, in the two 
equations for i=o and obtain two equations of the form 



i*'o + Wo') + <Vo(eo')=o 1 



(l6) 



in which everything is known except h . If the work be correct these 

should give the same value of A 0) and thence, substituting in (15), the values 
of A,, x' . 

The coefficients (/),-, (/),■, (c) t , (c') t being independent of x, are found once 
for all ; they are given below. The advantage of this procedure arises from 
the omission of the equations for i=o in the solution of the linear equations : 
the approximations are rapid. 

The same process is used for the nodal motion with the c-equation. 
But as we here put x =o = /\ ' (§ 26). the equations (15), (16) reduce to 



h=i\ + gx*(ff);, (except i=o) 

l\ + gxlk($/o) = ••• • 



(is)' 
(16)' 



The equation (16)' serves then as a test. In all coefficients arising in 
the --equation we have q i =—q'^ ; . 

74. The following are the numerical results required for the method of 
this Section : — 



•■. 


(/);• 


(f% 


3 


— •00000 009 


— .00000 039 


2 


— 'ooooi 269 


— •00006 201 


1 


— •00213 J86 


— '01412 29 









— T 


+ '01261 31 


— •00293 22 4 


— 2 


+ '00002 655 


— 'ooooo 071 


—3 


— •00000 001 


— '01711 83 


Sum . . . 


+ •01049 S° 



(/o) =+6-97857 932 
(/'o)= + 2*34613 680 
(e ) = + 1-07940 266 
(c'o) =+ .02014 43 8 



Royal Astros. Soc. Vol. LIII. 



B B 



174 



Mr. Ernest W. Brown, Theory of the 



i. 


(0)i. 


(0% 


(3)i- 


4 




— "ooooo 007 






3 


— 'OOOOO 002 


— -ooooo 936 


- 


•ooooo 001 


2 


— 'ooooo 348 


— '00135 001 


— 


•ooooo 298 


I 

O 

— I 


— -00071 843 


- -22311 94 


— 


•ooiii 579 


+ -71453 80 


+ -00038 928 


+ 


•I9835 12 


— 2 


-j- -ooooi 970 


4- -ooooo 037 


+ 


■00030 486 


-3 


— 'OOOOO 066 




+ 


•ooooo 118 


-4 


— -ooooo 001 




+ 


•ooooo 001 


Sum 


+ 71383 Si 


— -22408 92 


+ 


•19753 8S 



Section (v). Values of A, u& YaX, i^/aA. 

75* The following tables show the characteristics, arguments, and types 
of coefficients of the terms of the third order according to the scheme adopted 
in Chap. I., Section (iv). The numerical results are given below in the same 
order, and will be found in the §§ given in the first columns. 



«' 


A 


Arguments. 


Types of Coefficients in %f-'/aA. 


76 


e 3 


2i±3C ; 2i + c 


(*'), (*") ; (A'), ( f£ ' 2 ) 


77 


eV 


2i±(2c + m) ; 2i±(20 — m) ; 2i±m 


(S-n), (e'V) ; (eW), («".») ■ («'„), («v) 


78 


ee' 2 


2J + (c+.2m); 2t±(c— 2m); 2'i±0 


( £J? 2 ), ( t y 2 ) ; ( £I / 2 ), ( t y) ; ( £Jp? '), (,' m ') 


79 


e /3 


2 ^±3 m > 2i±m 


(v 3 ), W 3 ) ', (vW), inn' % ) 


80 


ek 2 


2i + (o + 2g) ; 2i + (c — 2g) ; 2i + 


{*&), (e'F 2 ) ; (ek'*), (t'¥) ; (tkkf), (e'kk') 


81 


e'k 2 


2i±(m + 2g); 2-i±(m — 2g); 2i + m 


(nk% {n'V*) ■ (r,k'*), (,'A*) ; („**'), (,/kk*) 


82 


e 2 a 


2i { dz2C ; 2-ij 


(A), (e' 2 «) ; («'„) 


83 


ee'a 


2i 1 ±(c4-m); 2-i 1 + : (c — m) 


(eijct), (t'j/a) ; ( f ij'a), ( £ 'qa) 


84 


e'' 2 a 


2'i 1 ±2m ; 21 y 


(?/ 2 a), (ij' 2 a) ; (ijj/'«) 


85 


k 2 a 


2l,±2g; 2%y 


(F«), (k' 9 -a) ; (kk'a) 


86 


ea 2 


2i + 


( f a 2 ), (,V) 


87 


/ 

e or 


2i±m 


(ij.i 2 ), (ij'a 2 ) 


88 


0. 


21 x 


(a') 



Motion of the Moon. 



175 



§ 


A 


Arguments. 


Types of Coefficients in f^/aA. 


89 


ke 2 


±( 2 i + g±2c); ±(2i + g) 


±(kS-), ±{kt>*); ±(i«') 


90 


kee' 


±{2i + g±(c + m)} ; ±{2i + g±(c-m)} 


+ (&*), ± (W); ±(^V), ± (&'*) 


91 


ke' 2 


+ (2i + g±2m); +(2* + g) 


+ (*,»), ±(*'/»); + (*„/) 


92 


k 3 


±(2i + 3 g); ±(2» + g) 


+ (*»); ±(£W) 


93 


kea 


±(2'h+g ± c) 


±(&£a), ±(&£'a) 


94 


ke'a 


±(2*i+g+m) 


±(^ a )' ±(^'i' a ) 


95 


ka 2 


±(2i+g) 


±(&a 2 ) 



The following long-period terms have been obtained with the required 
accuracy by the method of Section (iii) : — 



Arguments. 
±( — 2 + 2C + m), 
±(— 2 + 2C — m), 

±m, 
±( — 1 +c — m), 



Coefficients. 

(«'„)„, («V)» 

(tlj'oi)-!, (fV) } 



The values of V x for these terms, obtained as shown in Section (ii), 
agreed, as far as they went, with the values obtained by Section (iii) ; this 
agreement furnished a valuable test. 

Equation (8) of Chap. I. was used to obtain (W)^ with sufficient 
accuracy. 

Preliminary values of the parts of the motion of the perigee and node 
having e 2 , e'% k 2 , as factors were given in an Appendix at the end of Chap. IV. 
The values found below differ slightly from these. This is partly due 
to the fact that they have been re-calculated by a different and more accurate 
method. In one case — that of g e = — an error was found in one of the final 
steps of the early calculations (where no test equation had been computed) ; 
this induced an error in c k , which was deduced from it by using the con- 
necting relation which I gave in a paper, "Investigations in the Lunar 
Theory."* These two quantities having been recalculated independently, 
and the values satisfying the relation just mentioned, they may be accepted 

as final. 

* Amer. Jour, Math., Vol. xvii. p. 349. 



i/ 6 Mr. Ernest W. Brown, Theory of the 

The numerical results now follow. The values of A K for the terms 
arising in u are given in two parts— those arising from the expansion of 

m%- T jr\ denoted by K K , and those arising from - a ® ' £-, denoted by 8 A . Then 

except for the exponents 21 ±c, 2^ ±g, where 

The numbers are the coefficients corresponding to the power of % (that is, 
the argument) which is placed at the head of each pair of columns. The 
separation of A k , B x in z is unnecessary, as the parts arising from 8 require 
but little calculation. 

The suffix of i x is omitted in the tables. Further details concerning the 
results will be found in Chaps. I.- IT. 



Motion of the Moon. 



177 



76. Characteristic e 3 . 
g , = o. Values of 







R e > = 


- -tie- 


K e „ = 


= B e ,. 




i. 


2i + 


3=- 


ii - 3c 


21 +C. 


2% — 


c. 


5 






+ •00000 615 


+ '00000 OI5 


+ '00000 


175 


4 


+ 


00000 


025 


+ •00025 °37 


+ •00000 883 


+ '00009 


*39 


3 


+ 


0000 1 


5 01 


+ •00618 376 


+ •00046 189 


+ •00371 


182 


2 


+ 


00079 


199 


+ ■00932 740 


+ ■01879 136 


+ •08978 


705 


1 


+ 


°3 3 5 I 


1864 


-•05285 4356 


+ •45337 5630 


+ '00093 


o745 





+ 


79860 


3673 


+ •09945 4882 


-•29234 3773 


-•09815 


4555 


— 1 


— 


18057 


2 435 


+ •00264 8901 


— •00031 9004 


+ "°5 8 5 1 


8714 


— 2 


+ 


00415 


328 


+ •00004 93 x 


+ ■01207 313 


+ •00155 


383 


-3 


+ 


00086 


99i 


+ •00000 074 


+ •00031 268 


+ -00002 


902 


-4 


+ 


00002 


152 




+ •00000 579 


+ 'OOOOO 


048 


-s 


+ 


00000 


040 




+ 'ooooo 007 


' 




Sum 


+ 


65639 


546 


+ •06506 716 


+ •19236 675 


+•05647 


029 











Values of 








i. 


(Or 


(*"> 


( 6V )>- 


(«% 


5 






+ •00000 


010 




+ "OOOOO 


002 


4 






+ 'OOOOO 


701 


+ - ooooo 009 


+ 'OOOOO 


138 


3 


+ '00000 


014 


+ '00038 


889 


+ •00000 678 


+ '00009 


907 


2 


+ ■00001 


122 


+ •00176 


65 


+ •00048 105 


+ •00528 


276 


1 


+ •00079 


516 


— ■00218 


478 


+ •02520 232 


— ■00568 


84 





+ ■04147 


214 


+ •00522 


993 


-•04231 894 


-•04231 


894 


— 1 


-•03023 


295 


+ '00004 


365 


+ •01685 68 


+ •00328 


278 


— 2 


+ •00184 


26 


+ 'OOOOO 


052 


+ •00068 577 


+ '00002 


537 


-3 


+ •00004 


5°9 


+ 'OOOOO 


001 


+ •00000 497 


+ 'OOOOO 


031 


-4 

— 5 


+ 'OOOOO 


032 






+ -ooooo 006 






Sum 


+ •01393 


37 


+ •00525 


18 


+ •00091 89 


-•03931 


56 j 



C e a = + '00268 571 



178 



Mr. Ernest ^Y. Browx, Theory of the 



J J. Characteristic eV. 
Values of K eV + S e v 



i. 






21+2C-1 


-m. 






2i — 2e — m. 




5 












+ "OOOOO 


711 






4 


— 


•00000 


O44 






+ •00033 


841 






3 


— 


'00002 


526 






+ •01192 


768 


— 'OOOOO 


024 


.2 


- 


'00120 


655 


— 'OOOOO 


006 


+ •21675 


309 


— •OOO02 


IOO 


1 


- 


•04218 


737 


— 'OOOOO 


553 2 


— '2II26 


on 


— •00007 


0187 





- 


74778 


991 


— '00046 


3*58 


— -13819 


518 


— •00047 


54" 


— 1 


— 


•27045 


446 


— •00077 


1829 


- '00445 


874 


+ '00046 


0625 


— 2 • 


+ 


•04960 


312 


-•00054 


348 


— '00009 


192 


+ 'OOOOO 


SSi 


-3 


+ 


•00141 


627 


— •00014 


.708 


— "OOOOO 


152 


+.•00000 


006 


-4 


+ 


■00002 


806 


— •00000 


166 


— "OOOOO 


003 






-s 


+ 


•00000 


046 


— 'OOOOO 


OC2 










Sum 


— ] 


•01061 


608 


-•00193 


282 


— •12498 


121 


— -oooio 


C64 



i. 






2* + 2C - 


-m. 






24- 


-2c + m. 




5 


+ 


'OOOOO 


004 






— 'OOOOO 


108 






4 


+ 


'OOOOO 


249 






— "00005 


063 






3 


+ 


•00013 


421 






— '00181 


981 


— 'OOOOO 


024 


2 


+ 


•00580 


739 


— 'OOOOO 


006 


-•03489 


248 


— '00002 


107 


I 


+ 


•16168 


213 


— 'OOOOO 


5509 


-•01655 


646 


— ■00012 


2352 





+ 


•91682 


513. 


— "00046 


0625 


+ •18214 


920 


+ •00208 


0639 


— I 


— 


•02119 


681 


+ •00047 


54" 


+ •01811 


441 


— "00322 


6913 


— 2 


— 


•00724 


120 


+ •00007 


019 


+ '00046 


244 


— •00003 


856 


-3 


— 


•00020 


x 93 


+ '00002 


IOO 


+ 'OOOOO 


837 


— "OOOOO 


040 


-4 


— 


'OOOOO 


395 


+ •00000 


024 


+ 'OOOOO 


013 






-5 


— 


'OOOOO 


009 














Sum 


+ 1 


•05580 


741 


+ •00010 


065 


+ ■14741 


409 


— •00132 


890 



Motion of the Moon. 



170 



Characteristic eV. 
Values of K eV + Q e v 



i. 


21 + m. 






21 — m. 




5 


— 'OOOOO 


020 






+ -ooooo 


109 






4 


— •0000 1 


021 






+ -00006 


185 






3 


— '00049 


120 


— 'OOOOO 


002 


+ "00284 


107 


— 'OOOOO 


002 


2 


-•01737 


829 


— 'OOOOO 


227 


+ -09173 


506 


— 'OOOOO 


226 


1 


— •32001 


82 2 


— '000 1 9 


2836 


+ 1-28586 


471 


— -ooo 1 9 


1967 





— -11424 


8ll 


+ '00014 


0471 


— -10877 


253 


+ -00072 


573° 


— 1 


+ •27689 


127 


+ -00449 


4856 


— -06274 


5 2 7 


— -00072 


5730 


— 2 


+ •01060 


170 


-•00134 


464 


- -00188 


121 


+ .00019 


197 


-3 


+ •00023 


060 


— 'OOOO I 


580 


— -00003 


788 


+ -ooooo 


226 


—4 


+ "OOOOO 


392 


— -ooooo 


016 


— 'OOOOO 


057 


+ '00000 


002 


-s 


+ 'OOOOO 


O06 






— -ooooo 


001 






Sum 


— -16441 


868 


+ '00307 


960 


+ 1-20706 


631 








Values of 



i. 


(e'l),. 


(«'Y 


>.■ 


(«V) ( . 


(A> 


5 






+ -ooooo 


008 






— 'OOOOO 


001 


4 






+ 'OOOOO 


691 


+ -ooooo 


002 


— 'OOOOO 


099 


3 


— 'OOOOO 


029 


+ ■00047 


850 


+ -ooooo 


156 


— '00006 


734 


2 


— '00002 


191 


+ ■02396 


981 


+ •000 1 1 


095 


—■00327 


SOS 


1 


-•00143 


289 


+ •03768 


72 


+ ■00590 


206 


+ '02902 


28 





-•06288 


748 


— -01411 


942 


+ •08337 


053 


+ •02431 


890 


— 1 


— -14741 


25 


— -00006 


809 


-■03817 


68 


+ •00036 


494 


2 


+ •00695 


O47 


— '00000 


TI2 


— '00113 


005 


+ 'OOOOO 


631 


3 


+ '00001 


7 r 4 


— -ooooo 


002 


— 'OOOOO 


207 


+ 'OOOOO 


007 


—4 

5 


+ 'OOOOO 


036 






— 'OOOOO 


004 






] Sum 


— •20478 


7i 


+ •04795 


39 


+ •05007 


62 


+ •05037 


16 



150 



Mr. EievEST ~W. Brows, Theory of the 



Characteristic eV, 
Values of 



i. 


("V'- 




(«Y)i. 


i 

1 


5 






+ 


•ooooo 


001 I 


4 


— -00000 


012 


+ 


•ooooo 


076 


3 


— -ooooo 


929 


+ 


•00005 


597 


2 


— -00062 


354 


4 


•00347 


999 


I 


- '02957 


776 


5 


•12638 


o33 


O 


- -19519 


75 


4 


•14774 


72 


— I 


+ '04554 


004 


— 


■00648 


982 


— 2 


+ -00017 


353 


— 


•00002 


5°4 


-3 


+ -ooooo 


307 


_ 


'OOOOO 


046 


-4 


+ -ooooo 


003 


- 


•00000 


001 


-s 










1 


Sum ... 


— -17969 


!S 


4 


•27115 


1 
39 i 



78. Characteristic ee' 2 . 
Values of K e ,„ + Q ee „. 



i. 






2*' + e- 


2m. 








21- 


-c — 2m. 




5 












+ 


•ooooo 


241 






4 


+ 


•ooooo 


017 






+ 


•00012 


est 






3 


+ 


•ooooo 


873 






+ 


■o°5 1 3 


044 


— 'OOOOO 


014 


2 


+ 


•00029 


198 


+ "OOOOO 


001 


+ 


•12386 


6 59 


— 'OOOOI 


320 


1 


+ 


•00371 


387 


— 'OOOOO 


380 


+ 


•09957 


946 


— '00 1 20 


073 





— 


•ioioi 


661 


— -00138 


S°3 


- 


•03198 


329 


+ -00293 


3 2 9 


— i 


+ 


■05016 


537 


+ -06034 


441 


+ 


•00053 


064 


— -00045 


811 


— 2 


+ 


•02162 


607 


— '01016 


166 


+ 


•00002 


797 


— -ooooo 


70s 


"3 


+ 


'00056 


850 


— "00010 


231 


+ 


•ooooo 


062 


— -ooooo 


008 1 


-4 


+ 


■0000 1 


°33 


— 'OOOOO 


°99 


+ 


•ooooo 


001 






~5 
Sum . 


+ 


■ooooo 


018 


— -ooooo 


001 






136 


+ -00125 


398 


_ 


•02463 


141 


+ '04869 


062 


+ 


•19728 



Motion of the Moon. 



181 



Characteristic ee' 2 . 
Values of K ee ,i + 8 K „. 



i. 






2(*+C — 2m. 










2J- 


- c + 2m. 




5 


+ 


•ooooo 


016 






+ 


•ooooo 


006 






4 


+ 


■ooooi 


003 






+ 


•ooooo 


193 






3 


+ 


•00049 


475 






+ 


■00006 


588 






2 


+ 


•01857 


417 - 


•ooooo 


°53 


+ 


•00106 


031 


— 'OOOOO 


069 


I 


+ 


•38487 


310 — 


•00004 


6i5 


- 


'01398 


063 


— '00028 


638 


O 


+ 


•22941 


999 


•00247 


794 


+ 


•08802 


983 


+ -01764 


269 


— I 


— 


•00540 


404 - 


•00018 


882 


+ 


■06570 


344 


- "02534 


819 


— 2 


+ 


•00015 


577 


•000 1 1 


935 


+ 


•00202 


853 


- -00037 


010 


-3 


+ 


•ooooo 


637 


•ooooo 


158 


+ 


•00003 


953 


— 'OOOOO 


403 


-4 


+ 


•ooooo 


014 — 


•ooooo 


002 


+ 


•ooooo 


063 


— "OOOOO 


004 


-5 












+ 


■ooooo 


001 






Sum 


+ 


•62813 


044 - 


•00283 


439 


+ 


•14294 


95° 


— '00836 


674 



i. 

5 




2i + 


0. 






22- 


-c. 




— 'OOOOO 


005 






— 'OOOOO 


073 






4 


— -ooooo 


310 






— '00003 


719 






3 


— -00016 


037 






- '00153 


051 


— 'OOOOO 


Oil 


2 


— -00628 


749 


— 'OOOOO 


043 


- -03870 


578 


— 'OOOOI 


092 


1 


— -14287 


981 


— '00003 


572 


- -14727 


365 


— '00093 


229 





- '17975 


583 


- -00139 


125 


— -06087 


008 


4- '01360 


no 


— 1 


- '05798 


850 


— '00761 


209 


- '02253 


804 


+ '01 000 


455 


— 2 


— -00621 


038 


+ -00299 


069 


— '00064 


816 


■f '00012 


062 - 


"3 


— '000 1 5 


985 


+ 'O0OO2 


956 


— 'OOOOI 


220 


+ 'OOOOO 


126 


-4 


— 'OOOOO 


283 


+ 'OOOOO 


029 


— -ooooo 


020 


+ 'OOOOO 


001 


-5 
Sum ... 


— -ooooo 


005 














- '39344 


826 


— -00601 


895 


— '27161 


654 


+ '02278 


422 



Royal Astrox. Soc, Vol. LIII. 



c c 



182 



Mr. Ernest W. Brown, Theory of the 



Characteristic ee /2 . 
Values of 



i. 


(«?% 


1 («V)«. 


(*V"-)i- 


(*'v% 


4 








+ 


•ooooo 


210 


+ 


•ooooo 


010 






3 


+ 


■ooooo 


on 


+ 


•00015 


848 


+ 


•ooooo 


759 


— 'OOOOO 


227 


2 


+ 


•ooooo 


579 


+ 


•00994 


322 


+ 


•00050 


707 


— '0005 1 


717 


I 


— 


•ooooo 


203 


+ 


•36932 


90 


+ 


•02381 


252 


— 'IOII2 


9i 


O 


— 


•05024 


49 


+ 


•10676 


52 


+ 


•14026 


79 


— '23325 


20 


— I 


— 


•78467 


7o 


— 


•0000 1 


322 


+ 


•29493 


28 


— "00039 


221 


— 2 


— 


•00147 


012 


+ 


•ooooo 


026 


+ 


•00003 


6 55 


4- "00003 


448 


-3 


+ 


•ooooo 


949 


+ 


•ooooo 


001 


— 


•ooooo 


018 


+ 'OOOOO 


047 


-4 


+ 


•ooooo 


012 


















r Sum ... 


— 


•83637 


85 


+ 


•48618 


51 


+ 


•45956 


44 


- "33525 


78 



i. 


(«?i')i- 


(«W)i. 


4 


- 


•00000 


003 


— 'ooooo 


°59 


3 


— 


•ooooo 


236 


— -00004 


352 


2 


— 


■00016 


176 


— -00264 


878 


1 


- 


•00797 


920 


- -08755 


90 





— 


•02023 


661 


— -02023 


661 


— 1 


+ 


•14228 


77 


+ '00040 


089 


— 2 


+ 


■00037 


254 


— "OOOOO 


957 


-3 


— 


•ooooo 


231 


— 'OOOOO 


014 


-4 


— 


'ooooo 


003 






Sum 


+ 


•11427 


79 


— '11009 


73 



• •03465 60 



Motion of the Moon. 



183 



79. Characteristic e n . 
Values of K e -, + Q ^,. 



i. 


2i+3tn. 




21- 


-3m. 




5 










+ •00000 


026 






4 


— -ooooo 


001 






+ •00001 


439 






3 


+ 'OOOOO 


003 






+ •00063 


617 


— -ooooo 


OOI 


2 


+ •00002 


943 


— "OOOOO 


001 


+ •01785 


243 


— -ooooo 


180 


i 


+ •00165 


270 


— -ooooo 


866 


+ ■04256 


833 


— ■00017 


494 





+ •02364 


0522 


-•00875 


9209 


+ '01902 


8519 


— ■01242 


3782 


— 1 


+ •00827 


230 


— •18242 


850 


+ •00027 


208 


— •00041 


049 


— 2 


+ •00292 


678 


-•00159 


328 


+ 'OOOOO 


321 


— •ooooo 


047 


-3 


+ -00006 


929 


— •0000 1 


486 






+ 'OOOOO 


OOI 


-4 


+ -ooooo 


116 


— 'OOOOO 


014 










-5 

_ 

Sum 


+ - ooooo 


002 














+ '03 6 59 


222 


— •19280 


466 


+ •08037 


539 


— •01301 


148 



i. 


2i + m. 




2s — 


m. 




s 


+ 'OOOOO 


OOI 






— "OOOOO 


010 






4 


+ -ooooo 


066 






— -ooooo 


629 






3 


+ -00002 


869 






— '00028 


158 


— '00000 


OOI 


2 


+ •00072 


679 


— 'OOOOO 


OOI 


— •00825 


586 


— 'OOOOO 


123 


1 


+ •00088 


954 


— •00002 


564 


— '04906 


880 


— '0001 1 


396 





— •02263 


5731 


-•00385 


1214 


— '02019 


3073 


-•00275 


2931 


— 1 


— -00919 


324 


+ •08674 


561 


+ '00009 


023 


— ■00087 


6 33 


— 2 


— •00124 


392 


+ •00067 


905 


+ •00010 


367 


— '00005 


386 


-3 


— -00002 


899 


+ 'OOOOO 


633 


+ 'OOOOO 


278 


— 'OOOOO 


064 


-4 

~5 


— -ooooo 


°5i 


+ 'OOOOO 


005 


+ 'OOOOO 


002 






Sum 


-•03145 


670 


+■08355 


418 


— •07760 


900 


-•00379 


896 



1 84 



Mr. Ernest AY. Brown, Theory of th 



ie 



Characteristic e' 3 . 



Values of 



i. 


(l 3 );. 




(i")« 




(rf-n')i 




(vn' 2 )i- 


4 






4- 'ooooo 


02I 


+ "OOOOO 


001 


— "OOOOO 


008 


3 






+ •00001 


620 


+ 'OOOOO 


056 


— •ooooo 


65s 


2 


— 'OOOOO 


004 


+ •00113 


404 


+ '00002 


762 


— "00044 


497 


I 


— '00003 


123 


+ •05909 


Il8 


+ "00021 


277 


—•02145 


800 


O 


-■04032 


334 


+ •06189 


440 


— •00900 


66 


— •00500 


46 


— I 


—•26598 


707 


— '00002 


034 


+ •08932 


148 


— '00064 


95° 


*— 2 


+ •00004 


936 






— 'OOOOI 


321 


+ 'OOOOO 


°35 


"3 


+ 'OOOOO 


132 






— •ooooo 


049 


+ 'OOOOO 


003 


—4 


4- 'ooooo 


001 




1 










Sum 


-•30629 


099 


+ ■12211 


569 


+ •08054 


21 


—•02756 


33 



80. Characteristic ek 2 . 
Sek a = °- Values of K A *=A ei *. 




+ 


2(— C+ 2g. 


'OOOOO 


001 


+ 


'OOOOO 


066 


+ 


•00004 


388 


+ 


•00236 


501 


+ 


•°7579 


4392 


— I 


'337" 


7389 


J_ 


• 5938 


0541 


+ 


•00187 


632 


+ 


•00003 


410 


+ 


'OOOOO 


049 


.— I 


'19762 


199 



Motion of the Moon. 



185 



Characteristic ek 



ii^—o. Values of 7f ck „=i? 6k =. 



?'. 


2/ + e. 




21 — C. 




5 






— 'OOOOO 


005 


4 


— 'O000O 


025 


— 'OOOOO 


3 r 3 


3 


— •0000 1 


523 


— •00015 


379 


2 


— "00072 


653 


—•00524 


052 


1 


-•02123 


4090 


-•09199 


9007 





-•03507 


4881: 


— '02732 


4124 


— 1 


-•04932 


6078 


+ '06866 


6563 


— 2 


+ •01296 


733 


+ •00145 


824 


-3 


+ '00028 


345 


+ ■00002 


175 


-4 


+ 'OOOOO 


425 


+ 'OOOOO 


028 


-5 


+ 'OOOOO 


007 






Sum 


-•09312 


201 


-•05457 


379 



Values of 



4 

3 

2 
1 
o 

— 1 

— 2 

3 

-4 

Sum 



(.#% 



+ •00000 027 

+ •00002 256 

+ •00138 166 

— -07120 080 

+ •00889 21 

+ •00005 33t 

+ -ooooo 030 

— ■06085 06 



(.'*«),. 



+ •00000 053 
+ •00004 989 



+ •003? 



}6 



-•01711 531 

+ •24987 870 

+ •00146 418 

+ •00001 008 

+ -ooooo 007 

+ •23814 17 



(<*' 2 X- 




(«'**){■ 


+ •00000 


002 






+ -ooooo 


085 


— 'OOOOO 


008 i 


— 'OOOOI 


633 


— "OOOOI 


355 i 


-•02889 


967 


-•00235 


066 


+ ■64704 


18 


-•46343 


59 


+•05538 


425 


+ •17077 


937 ! 


+ '00030 


609 


+ •00029 


727 ' 


+ 'OOOOO 


207 


+ 'OOOOO 


136 i 



+ 'OOOOO 002 



•67381 91 



+ ■00000 002 



— "29472 22 



i86 



Mr. Ebnest TV. Brown, Theory of the 



Characteristic ek 2 . 
Values of 



i. 


(etf) ; . 


(«'**■),. 


4 




— 'OOOOO 012 


3 


— •ooooo 041 


— -00001 274 


2 


— -00003 797 


— •00130 683 j 


r 


— •00279 137 


—•13063 320 1 


o 


— '01298 640 


— •01298 640 1 


— i 


+ •29872 990 


+ ■01621 815 


— 2 


+ -00369 243 


+ •00008 801 


-3 


+ •00001 816 


+ "ooooo 059 


-4 


+ •00000 012 




Sum 


+ •28662 446 

1 


-•12863 2 S4 



e k ,= + -05385 595 



81. Characteristic e'k 2 . 
Values of K eV + q eV . 



i. 




2i + m + 


2g. 






28— m- 


-2g. 




5 










+ 'OOOOO 


015 






4 


— •ooooo 


001 






+ 'OOOOO 


870 






3 


— -ooooo 


°5* 






+ '00043 


519 


— 'OOOOO 


001 


2 


— 'OC00I 


838 






+ •01232 


953 


— 'OOOOO 


099 


1 


+ '00038 


7658 


— 'OOOOO 


0271 


— '24500 


7602 


— '00020 


8242 





+ •09673 


8490 


— •00005 


9900 


+ '09029 


2445 


-•00529 


7319 


— 1 


-•29235 


4798 


-•03343 


2 7S3 


— '00832 


9223 


+ 'OOOOO 


0725 


— 2 


+ '00803 


763 


— •00148 


971 


— •00018 


592 


+ 'OOOOO 


004 


-3~ 


+ •00017 


185 


— -ooooo 


704 


— 'OOOOO 


278 






—4 


+ 'OOOOO 


255 


— -ooooo 


005 


— 'OOOOO 


004 






~5 
Sum j 


+ "OOOOO 


003 




- 










-•18703 


549 


-•03498 


972 


-•15045 


955 


-•00550 


580 



Motion of the Moon. 



187 



Characteristic e f k 2 . 
Values of K rV + Q eV . 



i. 




2« + m- 


-2g. 






24- 


m + 2g. 




s 


— 'OOOOO 


002 














4 


— 'OOOOO 


210 






+ 'OOOOO 


012 






3 


— '00011 


304 


— 'OOOOO 


001 


+ 'OOOOO 


679 






2 


-•00357 


5 2 3 


— 'OOOOO 


105 


+ -00035 


223 






1 


+ •10326 


8572 


— "00024 


0255 


+ •01107 


0715 


— 'OOOOO 


0040 





-•12143 


6400 


— '00164 


8838 


— '11412 


7077 


— 'OOOOO 


0725 


— 1 


+ '06401 


3724 


— '00006 


4252 


+ •10878 


3159 


+ •00529 


73 x 9 


— 2 


+ ■00135 


509 


— 'OOOOO 


051 


- '00344 


466 


+ •00020 


824 


-3 


+ •00002 


016 






— '00005 


398 


+ 'OOOOO 


099 


— 4 

-s 


+ 'OOOOO 


028 






— 'OOOOO 


067 


+ 'OOOOO 


001 


Sum 


+•04353 


104 


-•00195 


492 


+ '00258 


663 


+ •00550 


S79 



i. 




2 i + m. 








22- 


-m. 




5 


















4 


— 'OOOOO 


037 






+ 'OOOOO 


015 








— '00003 


183 






+ '00003 


380 






2 


— •00228 


182 


+ 'OOOOO 


013 


+ •00379 


855 


+ 'OOOOO 


014 


I 


— '11219 


5215 


+ •00001 


8123 


+ '24359 


2093 


+ '00002 


1752 





+ •01112 


0432 


+ •00210 


9609 


+ •01085 


3449 


+•00530 


5862 


— I 


+ •24707 


3790 


+ •03394 


4779 


— -11769 


1816 


-•00530 


5862 


— 2 


+ '00642 


043 


+ •00014 


863 


—•00193 


573 


— '00002 


175 


-3 


+ '00010 


319 


+ 'OOOOO 


09s 


— '00002 


555 


— 'OOOOO 


014 


-4 


+ 'OOOOO 


141 


+ 'OOOOO 


001 


— 'OOOOO 


031 






-5 


+ 'OOOOO 


002 














Sum 


+ •15021 


004 


+ •03622 


223 


+ ■13862 


464 








i88 



Mr. Erxest ^Y. Buowx, Theory of the 



Characteristic e'k*. 



Values o£ 



i. 


(i*% 


(v'k'% 


016 


(**"% 


<y<t 2 >. 1 


4 






+ 'OOOOO 


— 'OOOOO 


002 






3 






+ '00001 


505 


— 'OOOOO 


389 


+ 'OOOOO 


001 


2 


— "OOOOO 


001 


+ •00143 


Soi 


— '00047 


166 


+ 'OOOOO 


116 


I 


+ -ooooo 


630 


+ •10274 


226 


-•07635 


55 


+ '00008 


516 


O 


+ •00163 


202 


+ •04278 


164 


— •08014 


891 


+ '00269 


177 


-I 


-•21446 


825 


— '00081 


"5 


+ •00666 


731 


+ •12718 


46 


— 2 


+ •00412 


476 


— 'OOOOO 


C73 


+ '00005 


120 


— •00186 


746 


-3 


' + '00001 


413 


— 'OOOOO 


006 


+ 'OOOOO 


039 


— 'OOOOO 


421 


-4 


+ 'OOOOO 


009 










— '00000 


004 


Sum 


— '20869 


096 


+ •14615 


618 


— •15026 


II 


+ •12809 


10 



i. 
4 


(yW)i 




0>7*') 










— 'OOOOO 


001 




3 


+ "OOOOO 


045 


— 'OOOOO 


270 




2 


+ '00004 


861 


— '00025 


922 




I 


+ -00488 


83s 


— •02176 


661 







+ •48307 


75 


-•52538 


39 




— I " 


+ •24918 


465 


— •08400 


495 




— 2 


+ '00065 


513 


— '00016 


048 




3 


+ 'OOOOO 


383 


— 'OOOOO 


081 




j —4 


+ 'OOOOO 


003 








! . Sum 


+•73785 


86 


-•63157 


87 


{ 



Motion of the Moon. 



189 



82. Characteristic e 2 a. 
Values of jT e =„+ s eV 



21", 






21 + 20, 




2i- 


-2a. 


9 i 


+ 


•00000 001 




— 


•00001 077 




7 ! 


+ 


•00000 074 




— 


'00082 168 


— 'ooooo 035 


5 1 


_ 


•00008 103 


— 'ooooo 009 


— 


•03612 906 


— '00002 740 


3 i 


- 


•01077 °5° 


— 'ooooo 799 


- 


•06573 304 


4- '00014 077 


1 1 


— 


56564 675 


— '00061 101 


+ 


•18304 866 


— '00221 738 


— 1 i 


+ 


•57575 "8 


+ '00081 292 


— 


•07199 098 


+ '00283 9 2 8 


J ; 


_ 


•02847 660 


- '00807 552 


— 


•OO053 429 


— '00306 906 [ 


— 5 ■ 


_ 


•00534 300 


+ '00064 9 I 9 


+ 


•00000 433 


— '00004 O0 3 


~7 | 


_ 


•00006 515 


— '00013 766 


+ 


'00000 021 


— 'ooooo 043 


-9 i 


— 


•00000 042 


— 'ooooo 173 








Sum j 


— 


•03463 152 


— "00737 189 


+ 


•00783 338 


— '00237 460 



9 


— 


•ooooo 031 


7 


— 


•00007 051 


S 


— 


•00628 907 


3 


— 


•29381 090 


1 


— 


•17424 233 


— 1 


- 


•06079 808 


-.1 


- 


•04075 79 1 


-5 


— 


•00043 4 2 6 


-7 


— 


•ooooo 129 


-9 


+ 


•ooooo 003 


Sum 


— 


•57640 463 



+ 



-f- 



00000 004 

00000 328 
00025 2 5° 
00173 425 

■00562 393 
00820 455 
00126 848 

00001 642 
ooooo 018 



+ 



00277 397 



Royal Astrgsf. Soc, Vol. LIII. 



d D 



I go 



Mr. Ernest V. Beown, Theory of the 



Characteristic e 2 a. 
Values of 



2J, 


(«H> 


l'"V)i- 


(«'«)*. 


9 




— 'ooooo 017 




7 


+ "OOOOO Q02 


— '00002 360 


— 'OOOOO 100 


5 


— - ooooo 099 


— '00247 899 


— -00016 334 


3 


— '00025 786 


- '06279 45 


— -01741 144 


i 


— -03029 187 


- -07046 35 


— -10960 58 


— i ' 


-h "12685 76 


— '00347 720 


+ '21027 9 


-3 


+ ""SSi 33 


— '00016 375 


— -00049 °°6 


-5 •■ 


— -oooio 571 


— 'ooooo 080 


— 00007 754 


-7 


— 'ooooo 946 


— 'OOOOO OOI 


— -ooooo 038 


-9 


— -ooooo 006 






Sum .... 


+ '21170 50 


- '13940 25 


+ '08252 9 



$%. Characteristic ee'a. 



Values of K Re , a + i 



2i 


2(' + c + m. 


28 — 0- 


-m. 




9 


+ 


•ooooo 05 








+ -ooooo 19 






7 


+ 


•00003 67 








— '00008 15 


— 


'OOOOO 021 


5 


+ 


'00250 00 


+ 


'OOOOO 


004 


— '01113 I 4 


— 


"ooooi 929 





+ 


•13660 87 


+ 


•ooooo 


161 


— '333 I2 49 


- 


•00150 657 


1 


+ 


4-65374 56 


- 


•00038 


845 


--•15457 19 


4- 


•01267 3 8 4 


j — J 


— 


•19743 34 


+ 


•00791 


743 


+ '93078 13 


+ 


•0081 1 527 


| ■ -3 


— 


•06923 50 


+ 


•09670 


148 


4- '01512 50 


+ 


•00880 652 


I -s 


— 


'00082 91 


- 


'OIOI2 


338 


+ -00018 99 


+ 


•00009 282 


i ~1 


+ 


'0000 1 15 


— 


'000 1 1 


510 


+ , 'OOOOO 21 


+ 


■ooooo 091 


1 -9 

j Sum 


+ 


■ooooo 04 


— 


•ooooo 


117 




+ 


•ooooo 001 


+ 


4'5 2 54° 59 


+ 


•09399 


246 


+ '44719 05 


+ 


•02816 330 



Motion of the Moon. 



191 



Characteristic eefa. 

Values of J^+ Q teV 



21. 






2i'+C- 


-m. 








2i- 


-e + m. 




9 


+ 


•00000 


06 








+ 


•00000 75 






7 


+ 


•0000 1 


84 


— • 


'OOOOO 


001 


+ 


•00050 84 


+ 


'OOOOO 001 


5 


+ 


•00022 


6l 


— 


•00000 


083 


+ 


•02742 74 


+ 


•00000 061 


3 


— 


•01956 


28 


— 


•00007 


iS5 


+ 


•90964 64 


- 


•00004 517 


1 


— 


'37493 


92 


— 


•00415 


929 


- 


•80846 72 


+ 


•00545 089 


— 1 


— 


•79391 


6l 


+ 


•00386 


559 


- 


•07537 94 


+ 


•01923 289 


-3 


+ 


•19032 


70 


— 


•01502 


54i 


+ 


•00025 7 1 


- 


•03283 987 


-5 


+ 


•003 1 1 


OS 


+ 


•00197 


73i 


+ 


•00010 73 


— 


•00044 785 


-7 


+ 


•00003 


94 


+ 


"0000 1 


999 


+ 


"OOOOO 28 


- 


•00000 485 


-9 


+ 


•00000 


04 


+ 


•00000 


020 


+ 


•00000 01 


- 


•00000 005 


Sum 


- 


•99469 


57 


- 


•01339 


400 


+ 


•0541 1 04 


— 


•00865 339 



Values of 



28. 


0<*)i. 


(«Y«)*- 


(«?'«)(• 


OV);. 


9 










+ 


•00000 01 


7 


+ 


■00000 05 


— 'OOOOO 16 


+ "ooooo 02 


+ 


■ooooo 96 


5 


+ 


•00004 4° 


— '00051 09 


+ -ooooo 73 


+ 


•00099 47 


3 


+ 


•00454 88 


- '05773 38 


— '00053 3 2 


+ 


•09020 60 


1 


+ 


•40017 48 


- -13184 7 


- -04137 23 


+ 


•32607 7 


— 1 


+ 


•01876 3 


+ '13596 59 


- 78645 2 


+ 


•01623 91 


-3 


4- 


•1 1064 27 


+ -00107 47 


+ '01255 04 


— 


'00345 75 


~5 


— 


■00122 87 


+ 00000 51 


+ -00025 44 


— 


"OOOOI ^ 


-7 
-9 




•00000 40 


+ -ooooo 01 


+ -OOOOO 12 






Sum 


+ 


•53294 I 


- 'o53°4 8 


- '81554 4 


+ 


•43005 6 



i9- 



Mr. Eenest W. Brows, Theory of the 



$4. Characteristic c' 2 o 
• Values of K^ a + 8 e ««. 



2i. 




28- 


2m. 






21- 


-2m. 




9 


— •00000 


030 






+ "OOOOO 


124 






7 


— •0000 1 


921 






+ '00003 


767 


— 'OOOOO 


002 


5 


— •00108 


806 


— "OOOOO 


on 


— •00018 


847 


— 'OOOOO 


286 


3 


^•04289 


260 


— 'OOOO I 


784 


-•08297 


640 


— '00026 


318 


1 


— •49418 


300 


- '00443 


201 


-•07733 


041 


—•01694 


340 


— 1 


— •02683 


885 


+ •00651 


l62 


-•15682 


286 


+ •00130 


863 


-3 


-•01458 


690 


— •20106 


340 


—•00657 


451 


+ •00001 


217 


-5 


-f - O0020 


815 


— •00201 


212 


— 'OOOIO 


781 


— '00001 


889 


-7 


+ 'OOOOO 


792 


— •0000 1 


913 


— 'OOOOO 


141 


— '00000 


025 


-9 


+ ■00000 


015 


— 'OOOOO 


Ol8 


— 'OOOOO 


002 






Sum 


-'57939 


270 


— •20103 


317 


-•32396 


298 


-•01590 


780 



2i 




28. 




9 


+ 'OOOOO 


l63 






7 


+ '00011 


559 






s 


+ •00672 


559 


+ 'OOOOO 


012 


3 


+ •25523 


830 


— '00001 


306 


I 


+ •50942 


997 


-■00515 


710 


— I 


+ '*79 r 3 


588 


—•00450 


279 


" — 3 


+ '0435 6 


335 


+ •06839 


549 


-s 


+ '00066 


997 


+ -00060 


357 


-7 


+ 'OOOOO 


796 


+ '00000 


560 


- -9 


+ 'QOOOO 


007 


+ 'OOOOO 


006 


Sum . . . 


+ •99488 


831 


+ •05933 


189 



Motion of the Moon. 



193 



Characteristic c /2 a. 
Values of 



2i. 


(ifa)i. 




(V"«)i. 




(irfa)i. 




7 


— "00000 


02 


+ 'OOOOO 


II 


+ "OOOOO 


15 




5 


— '00002 


75 


+ - oooo6 


42 


+ '00015 


08 




3 


— '00236 


65 


+ ■00181 


74 


+ •01214 


19 




1 


~-I0866 


00 


+ •07247 


89 


+ •01287 


7 




— 1 


-■22746 


69 


+ '06296 


01 


+ •24960 


2 




— 3 


-■05932 


75 


— •00033 


14 


+ •02080 


95 




— 5 


— "OOOIO 


54 


— "OOOOO 


3i 


+ '00004 


58 




-7 


— '00000 


°3 






+ "OOOOO 


02 


Sum ... 


"•39795 


43 


+ •13698 


72 


+ ■29562 


9 



8 z . Characteristic k 2 c 



Values of E Vli + Q, ■ k? „. 



2i. 




2S+2g. 






24- 


-2g. 




9 










— "OOOOO 


019 






7 


— 'OOOOO 


039 






— "00001 


583 


— "OOOOO 


001 





— -00003 


127 






— '00107 


524 


— 'OOOOO 


139 


3 


— -00219 


2 73 


— 'OOOOO 


028 


— •02025 


402 


— '00023 


440 


1 


— -10174 


53° 2 


— '00005 


1640 


+ •07691 


4545 


-—■00386 


8083 


— 1 


4- '31567 


8728 


— •01402 


7700 


-•30052 


2 743 


-•01234 


2006 


-3 


— •00281 


851 


-•02319 


391 


— '00124 


397 


— '00008 


150 


-5 


— •00082 


686 


— '001 1 1 


281 


+ '00001 


569 


— "OOOOO 


060 


-7 


— 'OOOOO 


723 


— 'OOOOO 


681 


+ 'OOOOO 


043 


— '00000 


001 


-9 
Sum 


— •ooooo 


007 


— 'OOOOO 


005 










+ '20805 


637 


-.•03839 


320 


— '24618 


133 


— •01652 


800 



194 



Mr. Ernest W. Bbown, Theory of the 



Characteristic k 2 a. 
Values of l' k% + S? k =„. 



2i 


21. 


9 


— 'OOOOO 


013 






7 


— -ooooo 


700 






S 


— -00034 


902 


+ '00000 


018 


3 


— •01038 


934 


+ '00002 


S77 


i 


+ •05314 


0982 


+ '00381 


1636 


— i 


+ •04282 


1667 


+ •02680 


3012 


-3 


— -03191 


SSi 


+ ■02342 


170 


-5 


— •00027 


549 


+ •00014 


068 


-7 


— 'OOOOO 


MS 


+ '00000 


098 


-9 






+ •00000 


001 


Sum ... 


+ ■05302 


471 


+ •05420 


397 



Values of 



28. 


(*%)«. 


(*'*<*)!. 


(U'a)i. 


9 






— 'OOOOO 


002 






7 






— 'OOOOO 


229 


+ -ooooo 


015 


5 


+ 'OOOOO 


Oil 


— •00037 


096 


+ '00002 


5°9 


3 


+ •00001 


162 


-•07251 


89 


+ -00430 


805 


1 


+ •00131 


775 


-•43498 


445 


+ '84123 


8x 


— 1 


+ •18884 


734 


-•06346 


523 


-2-33191 


84 


-3 


+ •16090 


08 


— •00005 


398 


— "00244 


631 


-5 


— •00041 


866 


+ -ooooo 


052 


— 'OOOOO 


465 


-7 


— 'OOOOO 


086 






+ 'OOOOO 


001 


-9 


— 'OOOOO 


001 










Sum ... 


+•35065 


81 


-'57139 


53 


—1-48879 


80 



Jlotion of the Moon. 



195 



86. Characteristic ear. 
Values of X ea * + 



*-&a- 1 CO Qa" 



i. 






21 + C. 




21 — c. 


5 










+ 'OOOOO 


05 






4 


+ "OOOOO 


24 






+ '00002 


45 


— 'OOOOO 


02 





+ '00012 


36 


— "OOOOO 


01 


+ •00117 


89 


— 'OOOOO 


22 


2 


+ "00520 


57 


— "OOOOI 


19 


+ •04732 


91 


— •00023 


56 


1 


+ •16389 


96 


— '00I2I 


33 


+ •07138 


99 


+ '00490 


20 





— '13821 


47 


+ •00035 


22 


— •O4822 


18 


+ •00736 


36 


— 1 


+ •02858 


°3 


+ •01208 


22 


+ '02221 


98 


+ ■00169 


14 


— 2 


+ •00750 


77 


+ '°3393 


88 


+ 'OO062 


41 


— •00922 


16 


-3 


+ •00012 


52 


— •00184 


24 


+ '00001 


36 


— '00008 


67 


—4 


+ 'OOOOO 


23 


— 'OOOOI 


63 


+ "OOOOO 


02 


— 'OOOOO 


08 


-5 


+ "OOOOO 


01 


— 'OOOOO 


01 










Sum... 


+•06723 


22 


+ •04328 


4i 


+ •09455 


88 


+ '00440 


99 

■ ■ 



Values of 



i. 


(« 2 );. 


(*'«% 


4 

3 

2 
1 


— 1 

— 2 
-3 
-4 


+ •00000 22 
+ •00015 7 1 
+ •00899 53 

— ■01666 28 

— •22632 4 
+ •00943 38 

— •oooii 98 

— - ooooo 04 


+ '00000 04 
+ •00003 JI 
+ •00229 1I 
+ •10151 I 

— "01666 28 
+ •00183 95 
-•00054 38 

— 'OOOOO 2 I 


Sum 


— •22451 9 


+ -08846 4 



C^= — -Q22I2 6. 



196 



Mr. Ernest W. Brown, Theory of the 



87. Characteristic e'a 2 
Values of K eW >. + Q eV . 



i. 


2«+ m. 




21 — 


m. 


I 
1 


5 












+ 


OOOOO 


02 






4 


+ 


•00000 


01 






+ 


OOOOO 


56 






3 


— 


•00008 


40 


— -ooooo 


01 


+ 


00013 


63 


— 'OOOOO 


07 


2 


— 


•01320 


15 


— •0000 1 


42 


- 


00223 


69 


— "00008 


17 


1 


— ] 


•06476 


55 


— •00261 


78 


+ 


04455 


94 


— '00852 


49 





+ 


■23300 


345 


"•00545 


494 


+ 


20737 


15s 


— •00450 


490 


—1 


+ 


•00948 


95 


— •01227 


27 


- 


21793 


97 


+ •00700 


S6 


— 2 


— 


•00075 


36 


-•08834 


34 


- 


00061 


S 2 


+ •01108 


38 


-3 


+ 


•00003 


33 


•—•00070 


66 


+ 


OOOOO 


45 


+ -00005 


95 


-4 

-5 


+ 


•00000 


08 


— 'OOOOO 


56 


+ 'OOOOO 


01 


+ 'OOOOO 


04 


Sum . , . 


— 


•83627 


74 


— • 1 0941 


53 


+ •03128 


59 


+ •00503 


7i 



Values of 



i, 
4 


{va-'H- 




(i'<Oi. 








+ -ooooo 


or 


3 


— •ooooo 


22 


+ •00001 


15 


2 


— •00051 


33 


+ "00063 


84 


1 


-•09673 


62 


+ •00542 


42 





+ •05209 


5 


+ •07246 


1 


__ 1 


—•01035 


45 


— '02641 


18 


— 2 


—•01043 


59 


+ •00131 


14 


"3 


— '00002 


64 


+ -ooooo 


27 


-4 


— -ooooo 


02 






Sum ... 


-•06597 


4 


+'05343 


8 



Motion of the Moon. 



197 



•88. Characteristic a 3 
Values of 



/O+S.- 



28. 


[ 

2i. j 


7 
5 
3 
1 

— 1 

! -» 

i ° 

I "5 
1 ~~ 7 


j 

— •00001 2 ! 

— •OOO94 2 — 'OOOOI O j 

— -01711 5 —-00141 9 
+ ■00371 9 —-00396 6 ; 
+ •00142 4 —-00726 8 

— •00256 8 --01393 l 1 
-•00013 -'°i353 6 

— ■00008 9 


Sum ... 


— •01562 4 --04021 9 



i 

21. | («"){. 


7 

S 

3 

1 

— 1 

-3 

-5 

1 ~ 7 


+ •00000 1 
+ ■00001 5 

— •00125 9 

— ■09620 
+ •27025 
-•00349 9 

— •00085 8 

— 'OOOOO 2 


j Sum ... 


~ ~ ' 

+ '16845 



8q. Characteristic k 3 . 









Values 


Df 








Ak". 




i? k ». 








i. 

5 








(**)* 




(W-'),-. 


2i + 3 





2J + g. 


c- 












4 






— 'OOOOO 010 






— •ooooo 00014 


3 


+ 'OOOOO 


007 


— •ooooo 744 


+ 'OOOOO 


00009 


— ■ooooo 0164 


2 


+ -ooooo 


543 


— •00049 643 


+ 'OOOOO 


OIII 


— "O0002 1515 


1 


+ -00036 


1540 


— •02324 4118 


+ '00001 


4145 


— •00277 8178 





+ •01675 


5 6 77 


— -01731 6970 


+ •00168 


5259 





— 1 


— •02777 


8633 


+ •00363 2982 


-•06974 


485 


-•01224 585 


— 2 


+ •00186 


956 


-— '00042 284 


— •00158 


665 


— 00007 8594 


-3 


— 'OOOOI 


059 


--•ooooo 958 


— 'OOOOO 


620 


— ■ooooo 0527 


-4 


— •00000 


048 


— 'ooooo 014 


— 'OOOOO 


0039 


— 'ooooo 0004 


-5 


— -ooooo 


001 




— 'OOOOO 


00002 




Sum . . . 


—■00879 


744 


— •03786 464 


— •06963 


822 


-•01512 483 



g v = — -00806 
Royal Astros. Soc, Vol. LIII. 



62 55- 



E E 



198 



Mr. Ernest W. Brown, Theory of the 



90. Characteristic ke 2 . 
Values of 



j. 




A 


<&• 




Bkt'- 




2i + g + 


2C. 


2J + g- 


2C. 


2i + g. 


s 






+ 'OOOOO 


036 


+ "OOOOO 


005 


4 


+ 'OOOOO 


012 


4 'O0O02 


174 


+ "00000 


360 


3 


+ 'OOOOO 


910 


+ - ooio6 


7Si 


+ •00022 


°53 


2 


+ '00056 


581 


+ •03177 


374 


+ •01086 


145 


1 


+ •02815 


3088 


—•00388 


7463 


+•32763 


8723 





+ •86762 


4162 


+ -00460 


5675 


+ •00651 


0052 


— 1 


-•01571 


9318 


— •02414 


0044 


+ ■00267 


1924 


— 2 


+ ■00148 


491 


— •00079 


507 


— "00902 


610 


-3 


— '00085 


55° 


— •0000 1 


611 


— "00030 


517 


-4 


— ■00002 


977 


— 'OOOOO 


025 


— "00000~ 


621 


-5 


— 'OOOOO 


063 






— "OOOOO 


OIQ 


Sum ... 


+ •88123 


197 


+ '00863 


009 


+•33856 


875 



Values of 



2, 


(*•*)*. 


<*«*- 


)<• 


(/be 


)»• 


5 




+ 'OOOOO 


00048 


+ -ooooo 


00004 


4 


+ •00000 000 10 


+ 'OOOOO 


0487 


+ -ooooo 


00463 


3 


+ •00000 01 131 


+ '00004 


8331 


+ -ooooo 


4718 


2 


+ •00001 1586 


+ •00427 


6263 


+ '00046 


0150 


1 


+ •00112 1329 


+ -01672 


370 


+ ■03928 


1120 





+ ■09379 8299 


— '08141 


572 







— 1 


— '04400 679 


— '00307 


9003 


— •00716 


888 


— 2 


— "00161 311 


— •00003 


4589 


— •00124 


5876 


-3 


-•00013 55°° 


— 'OOOOO 


0350 


— "OOOOI 


4005 


-4 


— •00000 1470 


— 'OOOOO 


00032 


— 'OOOOO 


0141 




— ■00000 0015 






— 'OOOOO 


00013 


Sum ... 


+ •04917 444 


- -06348 


088 


+ •03131 


713 



• es = +-00318 6183, 



Motion of the Moon. 



199 



91. Characteristic keef. 
Values of -4 kee '- 



i. 


2t + g + C - 


•-m. 


2(' + g— C- 


-m. 


2{' + g + e- 


-m. 


2i + g — e + m. 


5 






+ -ooooo 


021 


+ "OOOOO 


001 


— "OOOOO 


003 


4 


— -ooooo 


014 


+ •00001 


399 


+ •00000 


096 


— "OOOOO 


205 


3 


— -ooooi 


008 


+ '00076 


2 53 


+ '00006 


148 


— •000 1 1 


203 


2 


-•OOOS4 


981 


+ •02945 


645 


+ •00318 


821 


- -00437 


856 


1 


— '02125 


360 


+ •39674 


479 


+ •10786 


825 


—•06183 


438 





-•28354 


837 


— •26071 


788 


+ •33223 


139 


+ •29819 


178 


— 1 


+•35289 


056 


+ •03417 


336 


-•03919 


252 


-•05967 


106 


— 2 


- -01531 


540 


+ •00126 


893 


+ •00654 


802 


-•00435 


546 


-3 


— •0OIO2 


615 


+ '00002 


634 


+ •00025 


093 


— 'OOOII 


193 


-4 


— "00002 


518 


+ "OOOOO 


041 


+ "OOOOO 


524 


— "OOOOO 


203 


S 


— 'OOOOO 


045 






+ "OOOOO 


009 


— "OOOOO 


003 


Sum 


+ •03116 


138 


+ •20172 


9 J 3 


+ •41096 


206 


+ •16772 


422 



Values of 



i. 


(J«Oi. 


0'„ 


)«. 


(W)i- 


(Mv) t . 


5 






+ 'OOOOO 


00023 


+ "OOOOO 


OOOOI 


— "OOOOO 


00003 


4 


— "OOOOO 


00014 


+ 'OOOOO 


02390 


+ "OOOOO 


00099 


— "OOOOO 


00336 


3 


— "OOOOO 


oiS79 


+ '00002 


3817 


+ "OOOOO 


09940 


— "OOOOO 


33°° 


2 


— "OOOOI 


5276 


+ •00220 


1482 


+ "00009 


2083 


— ■00029 


9393 


1 


— "00132 


549* 


+ •15623 


236 


+ •00707 


5760 


— •02027 


447 





-■07509 


456 


+ ■22035 


892 


+ •10625 


739 


-•25455 


973 


— 1 


-■31367 


703 


+ "01194 


449 


+ •03225 


857 


-•02563 


651 


— 2 


— "01 000 


189 


+ "00009 


4212 


+ ■00276 


085 


- -00033 


4698 


-3 


— "00009 


1842 


+ "OOOOO 


0822 


+ -00OO2 


0390 


— -ooooo 


3552 


-4 


— "OOOOO 


0874 


+ "OOOOO 


00070 


+ 'OOOOO 


0174 


— -ooooo 


00351 


-5 


— "OOOOO 


00083 






+ "OOOOO 


0002 


— -ooooo 


00003 


Sum 


— "40020 


7i3 


+•39085 


635 


+ ■14846 


622 


-•301 1 1 


172 



200 



Mr. Ernest W. Browx, Theory of the 



92. Characteristic ke' 2 . 
Values of 





1. 




A 


ke' - -- 


Bu.. 




2i + g + 


2m. 


2( + g— 2m. 


2i + g. 




5 






+ 'OOOOO 002 








4 


+ "OOOOO 


003 


+ '00000 220 


— 'OOOOO 


061 




3 


+ 'OOOOO 


163 


+ '00012 938 


— "00003 


696 




2 


+ '00005 


277 


+ ■00598 953 


— •00170 


201 




1 


— •00018 


°37 


+ •15607 842 


-•04382 


8741 







+ "00147 


449 


— '00328 213 


+•01075 


1875 




— 1 


+ '15457 


297 


-■00057 354 


-•04205 


6780 




—2 


— '00623 


77i 


-•00027 3i3 


+ •00314 


3!4 




3 


— '00023 


118 


— •00000 738 


+ 00009 


501 




-4 


— 'OOOOO 


469 


, — - ooooo 014 


+ 'OOOOO 


183 




-5 


— 'OOOOO 


009 




+ 'OOOOO 


002 


Sum . . . 


+ •14944 


78s 


+ •15806 323 


-■07363 


3 2 3 



Values of 



«'. 


(for);- 


(*y% 


(l- V r, 


);• 


5 






+ 'OOOOO 


00002 


— 'OOOOO 


OOOOI 


4 


+ "OOOOO 


00004 


+ 'OOOOO 


00293 


— "OOOOO 


00079 i 


3 


+ "OOOOO 


0032 


+ 'OOOOO 


2898 


— "OOOOO 


0786 


2 


+ "OOOOO 


1954 


+ •00027 


I35 2 


— '00007 


1657 


1 


— "00002 


8799 


+ •02119 


6537 


—•00525 


9644 





— '00455 


928 


+ •00832 


47 







— 1 


—•25292 


577 


+ •02434 


3°4 


+ •12438 


368 


— 2 


— •00147 


2986 


+ 'OOOOO 


4232 


+ -00064 


3737 


-3 


— 'OOOOI 


3*5° 


— 'OOOOO 


oi73 


+ 'OOOOO 


5169 


-4 


— 'ooooo 


0120 


— "OOOOO 


0002 


+ 'OOOOO 


0046 


~5 


— '00000 


00013 




26 


+ 'OOOOO 


00003 


Sum . . . 


-•25899 


812 


+ •05414 


+ •11970 


054 



g«,*= +-00564 6 



:>■>;>• 



Motion of the Moon. 



20C 



Characteristic kea. 



Values of 



21. 




A 


ea- 




(/reaV. 


(/t€'o)j. 


2( + g- 


-c. 


2( + g- 


-c. 


9 


+ 'OOOOO 


OOI 


+ 'OOOOO 


015 


+ "OOOOO 


OOOOI 


+ '00000 


I 

00015 I 


7 


+ 'OOOOO 


I0 S 


— 'OOOOO 


684 


+ 'OOOOO 


001 1 7 


— 'OOOOO 


0196 


5 


+ 'OOOOi 


277 


— '00141 


686 


+ "OOOOO 


0096 


— '00006 


6448 


3 


-•00397 


241 


—•10868 


994 


— 'OOOI7 


8806 


-■01376 


9493 ; 


i 


-■40314 


29S 


4- ; ooo8o 


657 


-•04580 


8867 


— '01 190 


80 


— i 


+ •00752 


301 


— •01874 


8Si 


+ •04850 


79 


+ ■09203 


77 


~3 


-•03088 


873 


+ •01689 


021 


+ -06482 


90 


+ "00233 


1866 j 


5 


+ '00382 


908 


+ •00023 


291 


+ '00067 


3777 


+ "OOOOI 


1563 ! 


_ 7 


+ '00006 


094 


+ 'OOOOO 


141 


+ 'OOOOO 


3486 


+ 'OOOOO 


0041 


-9 
Sum 


+ 'OOOOO 


055 


— 'OOOOO 


002 


+ 'OOOOO 


0017 


— "OOOOO 


00003 


-•42657 


671 


— -11093 


092 


+ '06802 


66 


+ •06863 


70 



94. Characteristic ke'a. 
Values of 



2i. 




A 


ke'a- 




(l-na),. 




(/.V«) 




2 i •■*- £t *+ 


m. 


2S + g- 


m. 


9 


+ "OOOOO 


007 


+ 'OOOOO 


° T 5 


+ '00000 00008 


+ 


•00000 


00015 


7 


+ "OOOOO 


686 


+ 'OOOOO 


797 


+ '00000 01232 


+ 


•00000 


01283 ; 


S 


+ "00057 


193 


+ "000 28 


744 


+ •00001 82526 


+ 


'OOOOO 


8209 I 


3 


+ '039 6 3 


137 


+ 'OO232 


832 


+ '00285 o:t 3 2 


+ 


'000 1 3 


6733 i 


) 


+ rSoi62 


I I I 


— "00466 


404 


+ •51292 301 


— 


•00839 


428 j 


— 1 


- -05655 


671 


+ I '79OI3 


231 


+ -04334 669 


— 1 


■51928 


973 j 


-3 


+ '03667 


959 


- -03946 


234 


+ ■01706 372 


— 


•02090 


548 ; 


— 5 


+ '0005 1 


284 


— "00099 


194 


+ •00005 4 2 99 


— 


•00009 


7841 


7 


— 'OOOOO 


°93 


— "OOOOI 


460 


+ '00000 0060 


— 


•00000 


0584 \ 


-9 
Sum 


— 'OCOOO 


014 


— "OOOOO 


016 


— '00000 000 1 3 


— 1 


■00000 


00037 j 


+ i'S2246 


599 


+ 1-74762 


311 


+ •57625 629 


•54854 


285 




















































202 Air. Ernest AV. Huowx, Theory of the Motion of the Moon. 



Values of 



II- . 



2 
i 
o 

— i 

— .: 

.■> 

—4 



+ "00004 5 7 

+ ■00228 18 

+ ■07204 28 

+ ' 0;! 497 59 

t'o-';^ 3.i 

— , c;c-7^ </) 

- '00003 63 



Sum 



+ ■'-5*. =5 .55 



4 

1 


— i 

— 2 

-3 

— 4 



Suia 






lPu-' ifu)il CM ;/■', /'"., U.S.A.: 
1899 January 21. 



■ 'a 1 . 



+ •00000 0009 

+ •00009 6^3 

+ -ooS62 452 

; — '=7777 o 

- -o-.'i 5?5 
■- ;ocoo 21 j 

— '00000 002 



— O&92S 5 



Errata in Part I. 

Page 45, line 1 1, for " a' " -read " a'." 
„ 63, last Vmc, for " ;] m 3 " read " ;; nr;'-' 3 ." 
„ 69, line 9, for " ; " read " , ". 
„ 75, line 4 from bottom, for " (. . .) " read " (. ,.) 9 ." 
„ 107, line 10, for " — '00585 014 " read " — -00585 0139." 
„ „ „ 11, for "— -01024 957 " read " — -01024 9560." 
» >. » 1 2, /or " +-02515 958" read" +-02515 9581." 

, 1 ;,./''/■ " + ooS.jS cip "/■•■'"'- —-c.-o'jS =914." 

,, 116, Appendix. See corrected values gi-en above. 



Theory of the Motion of the Moon; containing a New Calculation of the Expressions for 
the Coordinates of the Moon in Terms of the Time. By Ernest W. Brown, 
Sc.D., F.R.S. 

[Received 1900 May 25 ; read 1900 June 8.] 



PART III. CHAPTER VI. 

In the first two parts of this Memoir, published under the same title in the Memoirs 
of tlie Royal Astronomical Society in 1897, 1899, the general theory and the numerical 
results, up to and inclusive of the third-order terms, have been given. This part 
contains the numerical results for the terms of the fourth order. 

The methods adopted are in general the same as those used for the third-order 
terms. Instead of finding the values of uju directly and then deducing those of u k . 
as in the third-order terms, I have found the values of u k directly. This change was 
found advisable when preparations were being made for the calculation of the fifth- 
order terms. It was seen that the n on -homogeneous equations (17), (18) of Chap. I. 
would involve much more calculation than the homogeneous equations (6), (7), (8) 
of the same chapter, and the latter require the results for u k and not those for uju a . 
In the former case u/r 3 has to be expanded to the fifth order, an enormous piece of 
work ; in the latter case we only require the calculation of such expressions as u 2 , 
uBs, &.C.. to the fifth order, and this has been so arranged as to require much less 
computation. The only other change from the methods of Chap. Y. is that mentioned 
in 671. 

As in the earlier work all the assistance I have had in performing the computa- 
tions has been rendered by Mr. Ira I. Sterner, A.M., and I take this opportunity of 
again expressing my obligations to him for the ability and accuracy with which he 
has conducted the work allotted to him.* I have also done a considerable amount of 
calculation myself, especially in the later portions of the work. 

* The expense of making the computations necessary to obtain the results given below has been met 
by a grant from the Government Grant Fund of the Royal Society. 

Royal Astrox. Soc, Vol. LIY. b 



Mr. Ehxest W. Brown, Theory of the 

The following is the table of contents of Chap. V. :— 

Section (i). A brief outline of the steps followed in the application of the general 
method to the terms of the fourth order. 

Section (ii). The final numerical results for the series a, 4 and for the co- 
efficients of all terms of the fourth order in u, z. 

Section (i). Formuke and Method of Procedure. 

96. The method employed is in general the same as ^\? a ^.^?t^ 
products in the third line of equation (i 7 ) and in the second line of (18) of Chap. 1. 
are formed by putting 

and choosing the part, of the forth order. Tho products in the fourth and fifth lines 
of (17) and in the third line of (18) are similarly obtained with 

The additional parts of (17), (18) in which we put 



„=»!! *> — •*!! 



are respectively 



The meanings of the various symbols have been explained in Chap. I. 

n i ^ r-i 1 nrp treated in like manner. 

The parts arising from fl, namely, ^ £ ,-5-37, are treatea in 

In the parts of equations (17), (18), 

£-! (Z> 2 + 2 m D) % w„, ' -IPtz* 

we substitute 



U.. = U» 



nnd nroceed as follows : in those parts of the arguments of u, z 2 which contain c, g 
^« the values of c, g to the second order (that is, we must retam the parte 
rf these two quantities which depend on rf, e% k", a«) ; when the operates J) B 
Z vTbeen performed the portions of the fourth order must be retamed. In all other 
operations the parts c , g» of c, g will be sufficient. 

The general procedure in performing the calculates and the methods of testing 
the results are in other respects the same as those explained in § 66. 



Motion of the Moon. 3 

97. The series finally obtained before proceeding to the solution of the linear 
differential equations are not, in the case of equation (17), the actual right-hand 
members, but series %, where 



AC,- 



The reasons for this have been stated in § 71 (the symbol Z is there denoted by 

In consequence of this change the series s„ u 2 , s 3 , e 4 (§§ 68-70) must be replaced 
by series %, % %, h, where 

(P=2,3>4)- ' 



U 2 = : | 3— w 2» *P— ~4~a~ S l> 



Further, as we shall find u^/sl directly, instead of finding uju, first, we use 
series U x , U s , U 4> U 4 instead of the series TJ l9 TI 2 , U 3 , IT 4 , where 









a 


p 


(p=i, 2, 3, 4). 


The 


values of these eight new series are given below. 

2 11... ! f »■ 


6,. 


4 


8j. 

+ -ooooo 


023 


— -ooooo 003 1 


— 'OOOOO 002 


+ - ooooo 003 


3 


+ •00002 


287 


— -ooooo 402 


— 'OOOOO 210 


+•00000 300 


2 


+ •00217 


032 


— •00048 522 


— •00020 726 


+ -00028 048 


1 


+ •17116 


620 


-•06341 S l6 


-•01843 390 


+•02151 153 





-■29502 


375 


+ ■87776 191 


—•87862 801 


-■81835 400 


— 1 


-•00233 


SS3 


+ •01093 391 


—•00581 778 


—•00739 195 


— 2 


— •0000 1 


9Si 


+ •0001 1 479 


— •00004 631 


— •00006 191 


-3 


— 'OOOOO 


017 


+ •00000 113 


— •ooooo 038 


— •ooooo 053 


-4 






+ •00000 001 






Sum 


— -12401 


934 


+ •82490 732 


-•90313 576 


-•80401 335 






t. 


u,. 




tt r 


u,. 


u 4 . 


4 






— 'ooooo 004 






3 


— -ooooo 


022 


'OOOOO 608 


+ 'OOOOO 021 


— •ooooo 032 


2 


— -00002 


94*3 


— -00089 987 


+ -00002 939 


— •00004 410 


1 


— •00428 


758 


— '16212 885 


+ '00454 712 


— •00648 148 





-•73409 


342 


+ 2-18355 94 


+ 1 


-•93146 358 


— 1 


+•43443 


130 


+ -00036 969 


+ -00869 575 


+ •03592 927 


— 2 


+ •00015 


197 


— -ooooo 196 


— -ooooo 049 


— •ooooo 147 


-3 

-4 


— -ooooo 


036 


— '00000 001 


— '00000 001 


— 'ooooo 003 


Sum 


-•30382 


777 


+ 2-02089 23 


+ 1-01327 197 


— •90206 171 



Mr. Erxest W. Brown, Theory of the 



Section (ii). Values of % u^' 1 /^ ; A, izj&%. 

98. The following tables show the characteristics, arguments, and types of 
coefficients of the terms of the fourth order according to the scheme adopted in 
Section (iv), Chap. I. The numerical results are given below in the same order, and 
will be found in the 66 given in the first columns. 



99 
100 

101 

102 

I03 

104 

105 

106 

107 
108 
109 

no 

III 

112 

113 

114 
115 
Il6 



117 
Il8 

119 



Arguments. 



Types of Coefficients in i^f-'/aA. 



ee J 

e 2 k 2 
ee'k 2 

e' 2 k 2 
k* 

3 a 

eVa 

ee' 2 a 

e' 3 a 
ek 2 a 

e'k 2 a 

eV 
e«'a 2 
k 2 a 2 



21+40 ; 2^+20 ; 2% 

2i±(3.c + m) ; 2-i±(3C-m) ; 2i±(o + m); 
2i±(c — m) 

2i±(2C + 2m) ; 2i±(2C-2tn); 

2i±2o; 2i±2m ; 2% 

2i±(c + 3m) ; 2i±(c — 3m) ; 2t±(c + m) ; 
2i±(c— m) 

2t±4m ; 2t + 2m ; 21 

2t±(2C4-2g); 24±(2C — 2g) ; 

2-i±2C ; 2*±2g ; 2% 

2-i±(c + m + 2g); 2*±(c+m-2g); 

2 i+(c— m + 2g) ; 2t±(c-m-2g); 

2i±(c + m); 2i±(c— m) 

2-i±(2m4-2g); 2i±(2m— 2g) ; 
2i±2m ; 2i±2g ; 21 

2t±4g; 2-i±2g; 21 

2*i±3c ; 2ii±c 

2i { ±(2C + m) ; 2i 1 ±(2C—m); 

2ii±(c + 2m); 2i 1 ±(c-2m)j 
2*1 ±c 
2» 1 ±3m ; 2i 1 ±m 
2i!±(c + 2g); 2ii±(o— 2g); 

2ij±(m + 2g); 2'i 1 ±(m-2g); 
2"ij+;m 

2i±2C ; 2% 

2i+(o + m); 2t±(c — m) 

2'i±2g ; 21 



ke 3 
keV 

kea' 2 



Arguments. 



2-i + g±3CJ 2* + g±c 

2-i + g±(2c + m) ; 2i + g±(2c— m) ; 
2-i + gim 

2^ + g±(c + 2m); 2t4-g±(c— 2m) ; 

2l + g±C 



(' 4 ),(*' 4 ); (*V),(«' 3 ); (A' 2 ) 
.(«*>»), («'¥) ; ( £ V), (*'¥ ; ( eV, ?)> ("""') ; 

(eh'r,*), («^) 

( t V),(«'V s );('V),(*'V); ' 
(* s w')> (*'V) ; («'rM«V 2 ) ; («'W) 

(^ 3 ), U'n' 3 ) ; H% (*V) ; toV), *W 2 ) ; 

(eW 2 ), (e'-?V) 

(v% W 4 ) ; (-N), (nv n ) ; («V 2 ) 

( £ 2 M'), ( t '*kk');(ez'k% (e t 'k>*) ; («'**') 

(£,&»), («'»/*") ;(«jA' 2 ),(€'fl'* s ); 

( £ V£ 2 ), (*'^' 2 ) ; ("j'A' 8 ), (*'»**); 

(,**»). (^' 2 ) ; (>^' 2 )> (^ 2 ) ; , 

(, S H') ; (rpkK) ; ( W 'F), (^'A;' 2 ) ; (tin'kkf) 
{¥), (k'*) ; (kW), (K") j («' 2 ) 

(«'<.), (t«a) ; ( t Va), («' 2 a) 

( t V),(e'Vo>; («V«).(''V); 

(«V)> ("V«) 

( £l? 2 «), ( £ y 2 a) ; ( £! ,' 2 «), («Va) J 
(,'a), (r,' 3 a) ; (,^'a), ( W "a) 

(.Pa), (^ ,2 ») ; (* *'M» (^ 2 «) ; 

( f tf«), (t'M'a) 

(»,Po), (^' 2 «) J (^ ,2 a), (i,'A»a) J 
(ykk'a), (t/kk'a) 

(A.*)/ (e'V); (e.'a 2 ) 

(«i,a»), ( £ 'V« 2 ) ; (^'« 2 ), (''"/a*) 

(Fa 2 ), (A;' 2 a 2 ) ; (M'a 2 ) 



Types of Coefficients in t^/aA.. 



(^), (A.' 3 ) ; (&V), (A«' s ) 

(*«»,), (*«' 2 "')-; .(*'>'). (*''*«»); 

(kei'tj), (kit't]') 
(kerf), (k- t ' n n ) ; (*^' 2 ) ; (^'»J 2 ) 5 

(W), ^''""j') 



Motion of the Moon. 



§ ] 


\ 


I20 


ke' 3 i 


121 


k*e 


12 2 


kV 


123 


ke 2 <x 


124 


kee'a 


125 


ke' 2 a 


126 


k 3 a ; 


12/ 


kea 2 | 


128 


ke'a 2 



Arguments. 



Types of Coefficients in i^/aA. 



2i + g±3m; 2i + g±m 

2t + 3g ±c 5 2i + g±c 

2-i + 3g±m: 2t + g±m 

2i l +g±2c; 2i t + g 

+ g±(c4-m); 2i,+g±(c- 

2i, + 3g±2m ; 2t,+g 

2Ji + 3gJ 2 h+g 

2i + g±C 

2i + g±m 



(&,'), (A,") ; {hf-n'), {knn") 
(Ft), (kV) ; (Vk' t ), (#AY) 
(k\), (*V) ; (kVc'r,), (¥kW) 

(F 2 a), (kt'°-a) ; (k-t'u) 
(ktrjo), (k.-'t]'n) ; (ke.r{a) t (ke'rjn) 

(*,»«), (WM; (*w'«) 

(Fa); (Wa) 
(La 2 ), (F'a 2 ) 



The coefficients of t-~ x /aX change their signs when the corresponding arguments 

change their signs. 

The following long-period terms have been obtained with the required accuracy 
by the method of Section (hi), Chap. Y. : — 

Arguments. Coefficients. 

+ ( — 2 + 2c), (A')-» ("' 3 )x 

±(— 2 + 2C — 2m), (t 2 lj' = )_„ (t'V), 
±(2C-2g), (rk'%, ( t "L') 

±(2 + 2m-2g), ('/*% (>)'*)_, 

The values of W for these terms, obtained by the method of Section (ii), Chap. V., 
no-reed, as far as they went, with the values obtained by Section (iii) ; this agreement 
furnished a valuable test. 

In the solution, of the linear differential equations those sets of terms with 
arguments 2i have no small divisors ; the continued approximation method was, 
therefore, employed for such terms instead of the method of Section (ii), Chap. V.* 
The continued approximation method was also employed in the solution of the 
differential equations for the terms with characteristics e ri a, eV, ee'a\ &V, and for all 

the terms in z. 

The numerical results now follow. The values of % K for the terms arising in 
u are given in two parts— those arising from the expansion of mEt 1 /^, denoted by ^, 

and those arising from - ( (I) 2 + 2 mD)^ + — - 1 j , denoted by l\. Then 



3 rt( 'o4 

£- = -4^-°-_ 1 xpart, characteristic A, in- 

3- u d 



M, 



.1** 



. 4V 
' 3 rt *o£' 

f(Z? s +2mD)2^ + ^U'- 1 

( x ' OS ) 



where A K is the right-hand member of equation (17), Chap. I. 

* The method for obtaining the results given in the section referred to has been published in the 
Camb. Phil. Trans, vol. xviii. pp. 94-106, under the title, " On the Solution of a Pair of Simultaneous 
Linear Differential Equations, which occur in the Lunar Theory." 



6 Mr. Ebnest W. Bbown, Theory of the 

In the case of z, we have 

where A x is the right-hand member of equation (18), Chap. I., K, is the part arising 
from the expansion of «/r», and L k is the part arising from the expansion ot 
, 3ft 

The numbers are the coefficients corresponding to the power of £ (that is, the 
argument) which is placed at the head of each column cr pair of columns. 

The suffix of i x is omitted in the tables. Further details concerning the results 
will be found in Chaps. I.-1V. 



Motion of the Moon 



99. Characteristic e 4 . 
Values of M + i. For arguments 2^ + 40, 21, 8=0. 



i. 


21 + 40. 


2J - 4c 


i 

i 


2i„ 


1 


5 






+ -0O003 


95 


+ "OOOOO 


28 


4 


+ - ooooo 


02 


+ -OOI03 


*3 


+ -00013 


62 


3 


+ •00001 


08 


+ •00204 


86 


+ •00587 


17 


2 


+ •00059 


44 


+ •00427 


24 


+ •14884 


07 


1 


+ •02594 


23 


-•03457 


94 


— •06923 


47 





+•67584 


28 


+ •05789 


84 


— '06099 


65 


— 1 


—•22643 


43 


+ •00198 


63 


— •00928 


89 


— 2 


+ •00669 


89 


+ '00004 


43 


+•01353 


65 


-3 


+ '00045 


68 


+ '00000 


07 


+ '00045 


9 1 


-4 


+ •00010 


24 






+ •00000 


99 


-s 


+ •00000 


34 






+ 'OOOOO 


01 


| Sum 


+ ■48321 


77 


! +-03274 


21 


+"02933 


^9 







2i t 2C. 






2t — 20. 




5 


+ 'OOOOO 


CI 






+ •00001 


79 






4 


+ 'OOOOO 


£6 






+ •00077 


93 






3 


+ 'C0046 


28 






+ •01582 


81 


+ •00000 


10 


! 2 


+ •02001 


44 


— 'OOOOO 


02 


+ •00431 


95 


+ •00015 


18 


1 


+•51237 


13 


— •00002 


62 


+ •00596 


01 


+ •00018 


42 





— -24760 


9i 


—•00371 


37 


— •03008 


47 


— •00041 


35 


— 1 


+ •00727 


5i 


+ '00098 


30 


+ ■04523 


32 


— 'OOOOO 


16 


— 2 


+ '00121 


42 


+ '00001 


06 


+ •00154 


64 






— ■? 


+ •00187 


28 






+ •00003 


45 






-4 


+ •00006 


20 






+ 'OOOOO 


04 






-5 


+ 'OOOOO 


13 




15 










j Sum 


+ •29567 


35 


—•00275 


+ •04763 


47 


— •00007 


81 



Mr. Ebkest W. Brown, Theory of the 











Characteristic e 


* # 


















Values of 












i. 


(* 4 )i' 


(*' 4 )/- 


(«'«% 


("")<• 




(«V«)i. 


4 






+■■00003 


9 | 




+ '00001 


9 


+ 'OOOOO 


24 


3 

2 






+ '00020 


-> 



+ 'OOOOO 


8 


+ ■00069 


4 


+ ■00013 


21 


4--OOOOI 


1 


+ -OOOI4 


3 


+ ■00043 


8 


+ '00034 


4 


+ -00489 


88 


I 


+ •00054 


4 


— •O0O77 


4 


+ •01597 


r> 


— •00172 


8 


— -00639 


9 


O 


4- '02003 


5 


+ -OOI45 


3 


— '02093 


6 


+ ■00573 


7 


-•01524 


5 


— I 


—•01647 





+ '00002 


5 


+ •00474 


6 


+ •00118 


3 


+ '00222 


8 


— 2 


4- '00217 









+ •00039 


2 


+ •00001 


9 


+ "00036 


45 


—3 


4- '00004 









+ •00005 


1 






+ 'OOOOO 


57 


-4 


4- 'ooooo 


3 






+ 'OOOOO 


2 






+ 'OOOOO 


01 


Sum 


+ ■00633 


3 


+ ■00108 


9 


+ •00067 


3 


+ -00626 


8 


— -01401 


2 



100. Characteristic eY. 

Values of M+2. 



1 i. I 






2s + 3e + m, 








2i — 3e — m. 




1 


5 
4 
3 

I 

1, 2 

1 

' 












+ •00007 


26 




! 




•ooooo 


05 






+ ■00276 


44 






— 


•00C02 


Si 






+ •05506 


64 


— -ooooo 


21 


- 


•00128 
■04820 
•91127 


53 
09 
06 


— 'OOOOO 

— •00023 


16 

27 


+ •05714 

— •12700 

— •10600 


34 
10 
69 


— •ooooo 

+ -00002 

— -00020 


99 
49 

08 


— 1 

— 2 
-3 


+ 

+ 


'44433 
•02661 
•00794 


98 

37 

42 


— •00003 
+ ■00007 

— -00006 


60 

60 

92 


— •00429 

— •000 10 

— •ooooo 


96 
63 

22 


+ -0002 3 
+ 'OOOOO 
+ 'OOOOO 


40 
OI 


-4 


+ 


•00027 


94 


— -QOOOI 


50 










-5 


+ 


•ocooo 


63 


— 'OOCOO 


02 




. 






Sum 


| - 


1-37027 


86 


— '00027 


87 


1 —-12236 


92 


+ '00004 


77 



Motion of the Moon. 



Characteristic eY 



Values of £ + £. 



i. 


2i+3c-m. 


2i'-3c + m. 




5 
4 










— -ooooi 


05 








+ "OOOOO 


25 




i 


— '00040 


3 1 








3 

: 2 
I 


— I 


+ '00013 
+ '00624 
+ -18642 
+ 1 '09809 
- 'i99 J 7 


46 
15 

58 
98 

10 


— 'OOOOO 

— •00023 
+ •00019 


16 

36 

86 


— •00782 
+ ■01156 

— •06276 

+ T3395 
+ •01566 


47 
09 
16 

42 

24 


— 'OOOOO 

— 'OOOOI 

— '00007 
+ ■00117 

— •00162 


20 

09 

40 
18 

22 




; — 2 

; -3 


+ -00505 

— '00101 


43 
90 


— •00002 

+ 'OOOOO 


3° 
96 


+ -00049 
+ ■00001 


22 
02 


— '00002 

— 'OOOOO 


77 

04 




: - 4 


— '00003 


82 


+ 'OOOOO 


22 


+ ■00000 


02 








i -5 


— 'OOOOO 


02 
















Sum 


+ 1-09573 


01 


— •00004 


78 


| +'09068 


02 


— ■00056 


54 


> 


















21 — c — m. 






1. 






21 + c + m. 














5 


„ 


•00000 


03 






+ •00001 


97 






4 


— 


•0000: 


53 






+ '00096 


49 






3 





•00077 


45 






+ ■03453 


1 5 


+ 'OOOOO 


01 


2 





'02927 


29 


— ■00000 


08 


+ •57798 


75 


+ 'OOOOO 


04 


1 





•57"9 


50 


— •00012 


14 


+ •09193 


66 


+ •00193 


3 2 








•63687 


77 


+ •00135 


94 


— '20284 


99 


+ •00026 


58 


— 1 


+ 


•0345 l 


56 


+ •00209 


42 


-•06986 


67 


— •00025 


82 


— 2 


+ 


•07900 


65 


+ '00021 


27 


— -00267 


69 


+ •00014 


10 


, -> 




+ 


•00332 


03 


— '00020 


68 


— •00006 


43 


+ •00000 


24 


-4 


+ 


•00008 


35 


— 'OOOOO 


34 


— 'OOOOO 


09 






-5 
Sum 


+ 


'OOOOO 


16 


— "OOOOO 


01 








. 


— 


•I2I20 


82 


+ •00333 


38 


+ •42998 


15 


+ •00208 


47 



Royal Astros. Soc, Vol. LIT 



IO 



Mr. Ernest W. Brown, Theory of the 



V 



Characteristic e 3 c y 



Values of Jt + &. 



i. 


214-c— m. 


2i — c + m. 


5 


+ "OOOOO 


17 






— ■ooooo 


33 






4 


+ -00008 


90 






— •00015 


73 






3 


+ '00426 


64 






— ■00593 


J S 


— -ooooo 


02 


2 


+ -14161 


61 


— 'OOOOO 


22 


— -11616 


23 


— -00003 


40 


I 


+ 178220 


15 


— '00030 


33 


— -03064 


08 


— •00029 


97 


O 


+ -20914 


40 


— •00135 


79 


+ •07494 


72 


— ■00056 


43 


— I 


- -01-55 


70 


— '00034 


43 


+ •23267 


55 


+ •00165 


22 


— 2 


— -01469 


83 


— -00003 


47 


+ •01289 


11 


— •00098 


79 


-3 


— '00055 


16 


+ '00002 


95 


+ '00035 


40 


— -ooooi 


68 


-4 


— 'OOOOI 


36 


+ 'OOOOO 


°S 


+ 'OOOOO 


77 


— -ooooo 


02 


-s 


— 'OOOOO 


04 






+ 'OOOOO 


02 






Sum 


+ 2'iio49 


78 


— '00201 


24 


+ •16798 


05 


—•00025 


09 



Values of 



i. | 

4 

3 

2 

1 ■ 


—1 
—2 

-3 

' —4 


1 


(c'V),. 


+ "OOOOO 2 

+ •0001 1 
+ '00445 3 

+'05193 I 

— •02861 3 

— •00166 

— '00004 7 


(«'*>»>«■ 


— •00001 3 

— '00040 9 
+ '00307 6 
-•00913 3 
+ •00576 7 
+ •00024 4 
+ 'ooooo 5 


— '00002 4 

— -oo 1 2 5 5 

— •O4042 4 

— -06117 4 
+ -00900 2 
+ "00036 
+ -ooooo 4 


+ -00008 7 

+ •00321 8 

+ -00936 8 

. --00590 5 

— •00396 9 

— -00006 1 

— -ooooo 1 


Sum 


-•09351 1 


+ •00273 7 


+ •02617 6 


— -00046 3 1 



Motion of the Moon. 



if 









Characteristic 


eV. 














Values of 










i. 


0V„).. 




(«'Y) 4 . 


(A'A- 


(ee' s 7)) 4 - 


4 






+•00002 


+ 'OOOOO 


1 


— -oocoo 


2 


3 


— -ooooi 


5 


+ •00101 2 


+ ■00008 


3 


— •00017 





2 


— •00080 





+ •03124 7 


+ •00380 





-•00593 


2 


I 


— -02711 


8 


+ ■00408 


+ ■09003 


4 


— •01970 


4 


O 


-•08633 





— •04425 8 


+ •01113 


1 


+ -07463 


4 


— I 


+ ■03812 


8 


-•00268 8 


+ •03997 


4 


+ •01114 


3 


— 2 

-3 


+ ■00390 

+ ■00004 


1 

9 


— -00003 8 

— 'OOOOO 1 


— -00060 

— 'OOOOO 



8 


+ ■00019 

+ 'OOOOO 


9 

4 


-4 


+ 'OOOOO 


1 




— 'OOOOO 


1 






Sum 


— ■07218 


4 


— •01062 6 


; +'i444i 


4 


+ ■06017 


2 



1 o 1 . Characteristic ere 1 
Values of £ + &. 



2,/2 















21 — 20- 


-2111. 




*. 




22 + 2C + 2m 
















s 

4 

3 












+ -00004 


49 














+ •00204 


02 






+ •00001 


55 






+ •06459 


°5 


— ■00000 


10 


+ •00057 


27 






+ •86143 


06 


— •O00l6 


58 


1 


+ •00793 


85 


+ 'OOOOO 


40 


-•47189 


56 


— •OOO28 


97 





-•37149 


07 


— '00042 


61 


—•07081 


97 


— •00 1 00 


5 1 
89 


— 1 


— •66500 


19 


— '00105 


94 


+ -00030 


3° 


— -00034 


— 2 


+ •20280 


05 


— •00298 


76 


+ -00004 


17 


— ■00000 


73 


-3 


+ •00849 


83 


-■00133 


65 


+ 'OOOOO 


TI 


— 'OOOOO 


01 


—4 


+ •00021 


90 


— •00002 


05 










-s 


+ 'OOOOO 


49 


— 'OOOOO 


°3 










Sum 


— •81644 


3 2 


— -00582 


64 


+ •38573 


67 


— •00181 


79 



12 



Mr. Ernest W. Brown, Theory of the 



Characteristic eV 2 . 



Values of M + %. 



1 * 


22 + 20 — 2111. 


22- 


i 

-20 + 2m. 1 


5 






— 'ooooo 05 




4 


+ '00001 24 




— 'ooooi 73 




3 


4- '00072 90 




— '00278 01 


+ -OOO00 02 


2 


+ '02854 II 


— -ooooo 02 


—•18735 97 


— •ooooi 29 


I 


+ '599°4 28 


— -00002 69 


+ ■14699 65 


— '00046 45 


O 


+ 1-26100 51 


— •00126 II 


+ •25718 37 


+ •00742 25 


— I 


+ "^oS 3° 


+ -00089 5° 


+ ■06791 35 


— - oi2i8 97 


— 2 


- '03928 35 


+ •00015 55 


+ •00269 °3 


— '00031 19 


3 


- '°°°52 55 


— -ooooi 86 


+ •00006 59 


— •ooooo 46 


—4 


— "ooooo 42 


— -ooooo 03 


+ 'boooo 07 


— •ooooo 01 


-5 


— "ooooo or 








Sum 


+ 1-98857 01 


— •00025 66 


+ •28469 30 


— •00556 10 



;' i 


2J + 2C. 


2t- 


-2C. 


5 






— 'OOOOI 20 




4 


— •ooooo 48 




-•OOO58 51 


— •ooooo 01 


3 


— •00024 76 




— •O1832 28 


— 'ooooi 42 


2 


-•01039 93 


+ •00000 27 


-•23064 49 


— •00208 94 


1 


— •25413 96 


+ •00031 50 


-•O2504 78 


— -00311 93 





-•54218 73 


+ •04731 79 


— •O9083 51 


+ •00876 11 


: — I . 


— •02036 16 


— '01141 91 


— "02701 79 


+ '°°555 46 


i —2 


-'04993 83 


— •00067 12 


— •00096 10 


+ •00010 91 


; -3 

1 


— •00225 J 6 


+ •00038 18 


— '00002 02 


+ •00002 15 


! ~ 4 


— •00006 08 


+ •00000 59 


— 'ooooo 05 


+ •00000 01 


-5 


— "ooooo 07 


+ •00000 01 






Sum 


-•87959 16 


+ '03593 3i 


-"39344 73 


+ •00920 34 



Motion of the Moon. 



13 



Characteristic eV 



Values of & + t 


















2i-2m. 






i. 




21 + 2m. 














5 










+ -ooooo 


79 














+ -00036 


03 






4 

3 

2 


4- -ooooo 


27 












+ "00002 


&Z 






+ '01554 


27 


— -ooooo 


01 

06 

89 


— -OI244 


90 


+ •00000 


16 


+ -41891 


21 


— -ooooi 


1 


-I-D3332 


56 


— ■00017 


67 


+ 3-40300 


52 


— •00103 





— -11065 


41 


+ •00096 


19 


- -10257 


34 


+ •00056 


05 


— 1 


+ -75°79 


48 


+ ■00658 


16 


— -20699 


71 


— -00117 


71 
26 


— 2 


+ -05144 


80 


- -00866 


61 


— -00256 


42 


--•00016 


— 3 


+ -OOI57 


54 


— -00016 


17 


— -ooooi 


78 


— -ooooo 


31 


—4 


+ -00003 


37 


— -ooooo 


22 


+ -ooooo 


03 






-5 


+ -ooooo 


08 















Sum 


- -35254 


48 


— •00146 


16 


j +3-52567 


60 


— •00183 


19 



i. 




si. 








5 


— 'OOOOO 


19 








4 


— -oooii 


21 








3 


— -00497 


96 


— -ooooo 


01 




2 


- •I4 I S5 


47 


— -ooooo 


92 




I 


_, -23587 


3 2 


— -00076 


02 







— -3976° 


68 


— -00044 


40 




! 

1 —I 


— -24832 


43 


+ •00439 


86 




i — 2 


— -01642 


02 


+ •00294 


3° 




1 _ 3 


— -00048 


60 


+ •00005 


04 




! -4 


— -ooooi 


16 


+ -ooooo 


07 




-5 


— -ooooo 


03 






_ 


! Sum 

i 


j —2-04537 


07 


+ ■00617 


92 





14 



Mr. Ernest W. Brown, Theory of the 



Characteristic eV 2 . 









^ 


/alues of 










«'. 


; OV)i. 




1 
1; 




(<V 2 ) 4 . 




| OV).. 




4 






+ '00005 


3 






1 — 'OOOOO 


I 


3 


-+ 'OOOOO 


2 


+ •00275 


9 


+ •00001 


2 


— "ooo 1 6 


I 


2 


+ "OOOOI 





+ •08685 


8 


+ •00062 


3 


— '01601 





I 


+ '00011 





+ •07421 


6 


+ •02043 


7 


— '14196 




O 


— •02678 


9 


— '00684 


1 


+ •10603 


8 


+ •03200 


2 


— I 


-•28583 


I 


— 'OOOOO 


3 


+ •21767 




+ ■00135 


S 


— 2 


+ •02348 


4 


+ "OOOOO 


1 


— •00396 


8 


+ "00003 


4 


-3 


+ '00009 


9 






— 'OOOOI 





+ 'OOOOO 


1 


-4 


+ 'OOOOO 


4 














Sum 


-•28891 


1 


+ •15704 


3 


+ •34080 




—•12474 





i. 


0V)«- 




I (6'W) . 




(«V)<. 


(«y% 


(«W)i- 


4 






— "OOOOI 


5 






+ "OOOOO 


7 


— "OOOOO 


20 


3 


— 'OOOOO 


4 


— •00072 


8 


— 'OOOOO 


3 


+ •00037 


8 


— 'OOOII 


78 


2 


— '0002 2 


4 


— '02068 


2 


— •00073 


6 


+ ■01603 


8 


-•00514 


67 


i 


— ■00815 


4 


+ ■05116 




-•07832 


4 


+ •30827 


9 


— '10410 


4 





-•03958 


2 


— "00131 





- -18186 




+ '16309 




—■09951 


9 


— 1 


—•05663 




—•00042 


6 


+ -I °559 


3 


-•02533 





— '02118 


2 


— 2 


— '00702 


8 


— "OOOO I 


1 


+ •00083 


1 


— '00004 


7 


— '00O2I 


77 


—3 | 


— '00002 


2 






+ '00O02 


1 






— 'OOOOO 


58 


— 4 


— "OOOOO 


2 














— 'OOOOO 


01 1 


Sum 


-•11165 




+ •02799 




-•I5448 




+ '46242 




— ■23029 


1 

s ! 



Motion of the Moon. 



15 



102. Characteristic ee n . 



Values of .ft +?. 



i. 




21 + C + 3m 








2t-e- 


~3 m - 




s 








1 


+ •00001 


11 






4 








1 


+ ■00057 


24 






3 


— 'OOOOO 


23 






+ •02308 


42 


— 'OOOOO 


02 


2 


+ •00001 


06 






+ •52094 


60 


— '00002 


74 


1 


+ •00353 


66 


+ •00002 


17 


+ •26017 


23 


— •00414 


10 





— •04803 


63 


— -00372 


49 


— •01662 


47 


+ -00308 


04 


— 1 


+ ■13629 


36 


+ •13899 


01 


+ '00047 


49 


- -00034 


45 ! 


— 2 


+ '09169 


96 


— '04090 


63 


+ 'OOOOO 


48 


+ -ooooo 


03 


-3 


+ '00306 


65 


— •00057 


78 


— 'OOOOO 


01 






-4 


+ '00006 


57 


— •00000 


7i 










-5 


+ 'OOOOO 


11 


— 'OOOOO 


01 










Sum 


+ •18663 


5i 


+ '09379 


56 


+ •78864 


09 


— •00143 


24 



i. 


21 + c — 3m. 




21- 


-e + 3111. 




5 
4 


+ "00004 


19 






+ 'OOOOO 


09 




\ 


3 


+ '00205 


95 






-j- 'OOOI2 


66 






2 


+ "07211 


99 


— 'OOOOO 


12 


+ •00832 


00 


+ 'OOOOO 


13 


! I 


+ 1-18317 


59 


— •00015 


04 


+ •10347 


72 


— '00023 


73 


O 


+ '27486 


46 


-■00493 


83 


+ ■11154 


53 


+ •02718 


3i 


— I 


+ '03165 


59 


— ■00125 


28 


-I -20318 


44 


-■06705 


07 


— 2 


+ '00139 


35 


-•00057 


45 


+ ■00898 


11 


— '00176 


45 


: —3 


+ -0OOO2 


18 


— - coooo 


18 


-i- - O0O22 


34 


— -CO0O2 


60 


— 4 


+ 'OOOOO 


02 






+ XOOOO 


45 


— 'OOOOO 


03 


Sum 


+ 1-56533 


32 


— '00691 


90 


+ •43586 


34 


— '04189 


44 



i6 



Air. Ernest W. Brown, Theory of the 



Characteristic ee n . 



Values of M + 2- 



i. 




2? + c + m. 






2b — ( 


— m. 




5 










— -ooooo 


44 






4 


+ 'OOOOO 


13 






— -00024 


16 




| 


3 


+ 'OOOIO 


29 






— -00971 


52 


— -ooooo 


34 i 


2 


4- -00302 


38 


— -ooooo 


20 


-•21574 


67 


— -00041 


10 | 


I 


4- -01409 


57 


— -00027 


39 


-•37164 


88 


— -02742 


72 


O 


-•I549S 


99 


— •01772 


85 


— -04966 


63 


+ '02712 


44 ; 


— I 


-•I594S 


22 


-•03853 


74 


4- -OOI45 


78 


— -00097 


°s 1 


— 2 


-■03562 


96 


+ •01719 


29 


4- -00029 


53 


— '00008 


42 | 


-3 


— 'OOI22 


46 


-f '00024 


26 


4- -ooooo 


94 


— 'ooooo 


14 ! 


-4 


— -00002 


66 


4- '-ooooo 


3° 










-5 


— -ooooo 


04 


4- 'ooooo 


01 










Sum 


— •33406 


96 


-•039 TO 


32 


-•64526 


°5 


— •00177 


33 



i. 




2i + C- 


-m. 






2I- 


-c + m. 




s 

4 


— -ooooi 


78 






4- "ooooi 


34 






3 


— -00094 


85 


4- 'ooooo 


01 


— 'OOCOI 


01 


4-'oooco 


°5 


2 


-•03389 


63 


4-'ooooi 


58 


-•03797 


42 


4- -00006 


02 


1 


-•56836 


53 


4-'ooi96 


08 


— -14814 


17 


4- -003 1 8 


13 





— '19411 


40 


4- -01863 


26 


-•06658 


97 


4- -02765 


6 s 


— 1 


— •05160 


44 


4- -0049 1 


80 


— •09074 


64 


+ 'o3547 


34 


— 2 


— •00689 


49 


4- -00272 


61 


— •00401 


73 


4- -00083 


57 


-3 


— -00005 


02 


— -ooooo 


95 


— -00009 


86 


+ •00001 


19 


-4 
-5 


4 -'ooooo 


or. 


— -ooooo 


03 


— -ooooo 


13 


+ -ooooo 


01 


Sum 


-•85589 


13 


4- '02824 


36 


-'3475 6 


59 


+ •06721 


96 | 



Motion of the Moon. 



*y 









Characteristic ae n . 












Values of 






i. 


(^ 3 )»- 


OV 3 );. 


(«)' S )i- 


OVJi. 


4 






+ •00001 4 


+ -ooooo 


I 




3 






+ ■00078 6 


+ -00004 


2 


+ -ooooo 4 


2 


— -ooooo 


I 


+ •03522 I ] +-002I2 


8 


+ •00037 4 


I 


— -00007 


8 


+ •78390 J +-066I3 


8 


+ •00357 


O 


— -04064 




+ -IIIII 1 +'I73 8 5 




-•28327 


— I 


— 1-52648 




— -OOOOI 2 j +-03836 




+ -00097 5 


— 2 


— -00492 


4 




—■00003 


4 


+ •00014 7 


-3 


+ -00004 


7 








+ -ooooo 3 


—4 


+ -ooooo 


1 










Sum 


— 1-57208 




+ •93102 +-28049 


— -27820 



i. 


(«jV)j. 


(eW 2 ),:. 


(fliA- 




(eW)i- 


4 






— - ooooo 6 








3 


+ -ooooo 


I 


— •00031 1 


— •0000 1 


9 


— 'ooooo 5 


2 


+ -00006 


4 


-•01336 8 


—•00094 


1 


— -00116 8 


1 


—•00045 


I 


— •28623 


— ■02764 


6 


+ •12094 





-■19733 




+ •54293 


+ •23524 




-■74298 


— 1 


+ "50408 




— •00002 3 


— •41624 




+ -00049 2 ! 


— 2 


+ •00184 


5 


+ 'ooooo 3 


— '00008 


8 


— -00005 7 j 


-3 


— 'OOOOI 


6 




— -ooooo 


1 


— -ooooo 1 1 


-4 


— 'OOOOO 


1 










Sum 


+ •30819 


+ ■24300 


— •20970 




— ■62278 



Royal Asteon. Soc, Vol. LIV. 



D 



Mr. Ernest W. Bbowx, Theory of the 



103. Characteristic e /1 
Values of M + %. 



i. 




21 + 4m. 






21— 4m. 


s 










+ 'OOOOO 


13 






4 










+ OOOO5 


52 






3 










+ •002 57 


71 






2 


4- "0000 1 


02 


— 'OOOOO 


01 


+ '07920 


62 


— 'OOOOO 


22 


1 


4-'ooi32 


73 


+ •00014 


6S 


+ •13198 


73 


—'00036 


II 





+ •03140 


86 


—•00886 


25 


+ •02357 


85 


—•02153 


32 


— 1 


4-'02695 


70 


-•39736 


75 


+ •00022 


85 


— •00052 


41 


— 2 


+ ■01319 


27 


-•00549 


20 


+ 'OOOOO 


28 


+ 'OOOOO 


07 


-3 


+ -00035 


92 


— -00006 


61 










-4 


+ -ooooo 


75 


— 'OOOOO 


08 










-s 


+ 'OOOOO 


01 














Sum 


+ •07326 


26 


— •41 164 


25 


+ •23763 


69 


— •02241 


99 



i. 


2.1 + 2m. 


-28 — 2m. 


5 


















4 










— •00003 


04 






3 


— 'OOOOO 


81 






— •00143 


50 






2 


— '00003 


41 


+ 'OOOOO 


°3' 


-•04376 


96 


— 'OOOOO 


10 


1 


— '00080 


80 


+ 'OOOOO 


32 


— •15494 


59 


— '00034 


21 





; — '02145 


3° 


— '00619 


14 


— •01687 


08 


+ '00132 


00 


— 1 


! — '03026 


82 


+ •24551 


88 


— 'O0020 


08 


+ ■00031 


30 


— 2 


! — '00693 


33 


+ •00294 


84 


— 'OOOOO 


49 


— 'OOOOO 


01 


-3 


1 — '00019 


14 


+ '00003 


64 


— 'OOOOO 


II 


+ 'OOOOO 


02 


— 4 
-5 ■ 


| — 'OOOOO 


35 


+ 'OOOOO 


04 








\ 


j Sum 


-•05969 


96 


+ •24231 


61 


-•21725 


85 


+ -00129 


00 



Motion of the Moon. 



19 





Characteristic e 14 . 






Values of £ + &. 




i. 


2i. 


5 






4 


+ -00000 53 




3 


4--ooo2i 42 




2 


+ •00536 69 +-ooooo 


10 


1 


+ •01053 21 — -0O004 


58 





— •01872 34 —-00267 


3 2 


— 1 


+ -00I69 56 — -0II29 


18 


— 2 


+ •00078 57 — -00037 


20 


-3 


+ •00002 50 — -ooooo 


57 


-4 


+ •00000 04 — '00000 


01 


-5 






Sum 


— '00009 82 —-01438 


76 



Values of 



(i 4 )f. 



3 

2 

1 

o 

— 1 

— 2 
-3 
Sum 



+ "00001 o 

— •02698 

— •61854 2 
+ •00019 2 

+ •00000 6 

-•64531 



(n'% 



+ -00007 5 

+ ■00398 6 

+ •14687 7 

+ 04747 

— "00019 2 

+ 'OOOOO I 

+ ■19822 



(»y)<. 



+ 'OOOOO I 

+ •00018 I 

+ •05634 

+ •26249 4 

— '00006 7 

— -ooooo 3 

+ •31895 



(Wk 


(nW"-)i. 


50 


— -00003 


8 


+ 'OOOOO 


— -00198 


9 


+ •00019 


44 j 


— •06580 





+ •00245 


9 i 


—•08045 




-•00536 


i 


— •00019 


8 


-•00885 


8 ! 






+ •00000 


37 i 



—•14848 



+ •00000 03 



-•01155 6 



20 



Mr. Eknest W. Brown, Theory of the 



104. Characteristic e 2 k 2 . 
Values of M + %, For arguments 2i±2c±2g, 21, g=o. 



i. 


| 21 + 2C + 2g. 


2J — 2C-2g. 


2J + 2C — 


2g- 


2i — 2C+2g, 


5 






+ 'OOOOO 


50 






— 'OOOOO 


07 


4 


— 'OOOOO 


01 


+ '00O2I 


12 


+ - O0000 


II 


+ 'OOOOO 


II 


3 


+ •00000 


02 


+ '00430 


10 


+ •00002 


24 


+ 'OOOOO 


41 


2 
1 



— 1 


+ '00007 
+ '00500 
+ •22308 

-•69815 


85 

53 
59 
36 


+ "02174 
— '18229 
+ I'I7697 

+ "03394 


09 

57 
36 
87 


-•OOI75 
— ■27258 

+ "53053 
-•I5877 


OO 

99 
65 
59 


—•00591 

-•86503 

+ "55 OI 5 
-•05478 


44 

82 

59 
89 


— 2 

-3 

-4 


+ '03225 
+ '00096 
+ -00006 


26 
80 
01 


+ '00062 

+ "OOOOO 


72 
89 


+ •04235 
+ '00128 
+ '00002 


33 

22 
42 


+ •00037 

+ '00002 
+ 'OOOOO 


20 

35 
10 


-5 


+ "OOOOO 


17 






+ 'OOOOO 


°3 






Sum 


—•43670 


14 


+ 1-05552 


08 


+ •14110 


42 


-•37518 


46 



*. 


28 + 2C. 


\ 


2S- 


-2C. 




5 










— "OOOOO 


14 






4 


— 'OOOOO 


04 






— -OOOIO 


32 


+ 'OOOOO 


01 


3 


— -00002 


44 






-•00525 


50 


+ '00002 


08 


2 


— -00123 


16 


— 'OOOOO 


45 


—•16771 


28 


+ '00304 


27 


1 


— -02814 


68 


—•00052 


60 


+ '04896 


69 


+ 'OO369 


3 1 





+ 1-68846 


77 


- "07456 


96 


+ •68013 


38 


— '00829 


12 


— 1 


+ "04541 


74 


+ '01971 


2 5 


+ "07005 


51 


— -OOOO3 


14 


— 2 


— '03106 


66 


+ •00021 


32 


+ •00187 


82 


— "OOOOO 


°5 


-3 


+ '00182 


12 


+ 'OOOOO 


°5 


+ '00003 


40 






—4 


+ '00006 


20 






+ 'OOOOO 


08 






, -5 


+ "OOOOO 


r 9 














Sum 


+ 1-67530 


04 


-•05517 


39 


+ •62799 


64 


— •00156 


64 














_ 



Motion of the Moon. 



21 



Characteristic e 2 k 2 . 
Values of M + %. 



i. 


2i + 2g. 




21- 


-2"". 




24. 


5 


+ '00000 


01 






+ 'OOOOO 


04 






— 'OOOOO 


04 


4 


+ 'OOOOO 


09 






+ •00002 


74 






— 'OOOOI 


39 


3 


+ 'OOOOI 


91 






+ •00068 


73 






— '00076 


69 


2 


+ -00099 


15 






-■03325 


97 


+ 'OOOOO 


13 


— -03247 


94 


I 


+ '01302 


76 


— 'OOOOO 


01 


+ -28101 


50 


+ '00076 


36 


- -57621 


43 


o 


-4-51169 


89 


— -00009 


89 


-•67308 


93 


— '01560 


86 


-1-44031 


63 


— I 


+ -35823 


7 1 


+ •00168 


86 


+ •44117 


02 


— '00003 


52 


- '05952 


65 


— 2 


— -00820 


20 


+ 'OOOOO 


54 


+ •01307 


25 


— 'OOOOO 


01 


+ -02225 


23 


-3 


+ '00035 


II 






+ •00024 


35 






+ -00067 


3i 


—4 


+ 'OOOOI 


18 






+ 'OOOOO 


49 






+ -ooooi 


20 


-5 


+ 'OOOOO 


02 






+ 'OOOOO 


01 






+ - ooooo 


01 


Sum 


— 4-14726 


IS 


+ 00159 


5° 


+ •02987 


2 3 


— •01487 


90 


-2-08638 


02 



Values of 



i. 


(«***)«■ 


U"-k'% 


(.'&'»),. 


(«'«**)«• 


4 

3 

2 
1 



— 1 

— 2 

-3 

—4 


+ •00002 4 
+ •00085 7 
— '04605 1 
+ ■01017 3 
+ '00009 1 
+ '00000 4 


+ •00001 
+ '00042 2 
+ •00109 5 
— •03189 

+ '09359 9 
+ •00112 

+ 'OOOOI I 


— ■00017 

— •02398 8 
-■07730 

— -0I02I 8 

+ ■00422 4 
+ -00004 8 
+ -ooooo 1 


— -ooooo 7 

— •00079 ° 
-•07175 9 

+ '34043 
-•00454 5 
+ ■00008 9 

+ 'OOOOO I 


Sum 


1 - '03490 2 


+ •06436 7 


— -10740 


+ •26342 



22 



Mr. Ebnest W. Bbow^ t , Theory of the 



Characteristic e 2 k 2 



Values of 



i. 
4 


(«'**'), 




{*>H:k'). 




("'k% 




(«'*")«■ 


(«'*£')«• 






— 'OOOOO 


4 






+ 'OOOOO 


I 


— 'ooooo 03 


3 


— 'OOOOO 


I 


— •00027 


1 






+ 'OOOOO 


7 


— 'oooo2 43 


2 


— •00003 


2 


— •01602 


8 


— '00003 


2 


-■00323 


2 


— '00151 42 


I 


— •00102 


5 


-•03306 




— '00370 


3 


— '02056 




-■05637 1 


O 


4--o88j3 


1 


+ •25316 





-'35724 


2 


+ •01475 


2 


- -36003 


— I 


+ •05520 




+ •00622 


2 


+ '16072 




+ •03914 


8 


+ "°3S25 8 


— 2 


+ •00101 


8 


+ '00006 


9 


— ■00146 


8 


+ '00045 


8 


+ ■00247 °4 


-3 


+ •00026 


3 


+ 'OOOOO 


i 


+ '00003 


8 


+ 'OOOOO 


5 


+ •00002 72 


-4 


+ 'OOOOO 


2 






+ 'OOOOO 


1 






+ 'ooooo 03 


Sum 


+ •14416 


+ '21009 


—•20169 


+ •03058 


— •38018 4 



105. Characteristic ee'k 2 . 
Values of & + %. 



i. 




2i + e + m + 


2g- 






2« — e- 


-m-2g. 




5 










+ 'OOOOO 


42 






4 


— "ooooo 


°5 






+ •00021 


14 






3 


— -ooooo 


08 






+ •00877 


60 






2 


— '00006 


92 






+ •16983 


13 


— '000Q2 


30 


1 


- -00356 


07 


+ 'OOOOO 


01 


-'34429 


60 


+ '000l6 


72 





- -08618 


70 


+ '00005 


01 


-'36413 


27 


— •OOI79 


20 


— 1 


~ ri 9953 


59 


-•00937 


10 


-■02535 


01 


+ 'OOOOO 


33 


— 2 


+ "07 743 


85 


+ '00064 


02 


— '00060 


82 


+ '00000 


01 


-3 


+ -00306 


6 9 


— '000 1 5 


20 


— •0000 1 


04 






-4 


+ • 00008 


06 


— 'OOOOO 


16 


— 'OOOOO 


03 






-5 


+ 'OOOOO 


20 














Sum 


— 1-20876 


61 


-•00883 


42 


-'55557 


48 


— '00164 


44 



Motion of the Moon. 



23 









Characteristic ee'k 2 . 














Values of §, + Z. 








i. 




2J + e + in- 


2g- 




21 — < 


; — m + 2g. 




5 








+ -ooooo 


08 






4 


— 'ooooo 


21 




+ '00000 


08 






3 


— 'ooo 1 8 


74 




+ •00008 


57 






2 


— '00927 


28 


— •00000 20 


+ -00432 


54 






1 


-■30774 


52 


+ •0002 7 64 


+ •09701 


39 


— '00002 


o7 





+ ■00314 


35 


— ■OIO50 06 


—•00505 


45 


+ -00286 


25 


— 1 


+ •70749 


5i 


+ ■01781 40 


-•10767 


95 


+ •00265 


37 


— 2 


+ ■03875 


60 


+ ■00005 33 


—•00459 


77 


— •00016 


85 


-3 


+ •00089 


18 


+ ■00000 02 


— •000 10 


12 


+ 'OOOOO 


02 


-4 


+ ■00001 


5i 




— 'OOOOO 


13 






-s 


+ '00000 


03 












Sum 


+ •43309 


43 


+ -00764 13 


— •01600 


76 


+ •00532 


72 



i. 




2J + C- 


-m + 2g. 






2(- 


-c + m — 2g. 




5 










— 'OOOOO 


04 




1 

1 


4 


— 'OOOOO 


°5 






— ■00004 


96 






3 


+ 'OOOOO 


80 






— '00252 


47 






2 


+ '00045 


8l 






— •06776 


15 


— •00002 


05 ! 


1 


+ -02115 


l6 






+ '09205 


61 


— "00026 


88 ; 





+ •21254 


26 


— '0000 2 


l6 


+ •47352 


52 


+ •00138 


82 


— 1 


+ •30194 


12 


+ '00179 


35 


+ •13623 


37 


— *oooo6 


5° j 


— 2 


— •02729 


45 


— 'OOOI4 


91 


+ •00367 


59 


— '00000 


08 


— 3 


— '00103 


08 


+ 'O0O02 


!5 


+ '00006 


67 






-4 


— '0000 2 


3 2 


+ 'OOOOO 


02 


+ 'OOOOO 


03 






~~ 5 


— 'OOOOO 


01 














Sum 


-^•50775 


24 


+ 'OO I 64 


45 


+ •63522 


17 


+ '00103 


3t 



2 4 



Mr. Ernest "W. Brown, Theory of the 



Characteristic e/k 2 



Values of M + 8. 



1. 




2i'+c — m- 


-2g- 






21- 


-c + m + 2g. 




s 


+ 'OOOOO 


03 














A 


+ - OOOOI 


31 






— 'OOOOO 


08 






3 


+ "00054 


61 






— "OOOOO 


82 






2 


+ •01389 


61 


— 'OOOOO 


20 


— •00027 


59 






I 


+ -4795 2 


76 


+ •00019 


55 


+ •00611 


99 


— 'OOOOO 


46 


O 


-'°3394 


87 


— •00262 


24 


-•17243 


46 


+ -00074 


49 


— I 


—•08746 


22 


-•00288 


95 


+ •18225 


78 


— •02624 


26 


— 2 


- "0053° 


01 


— 'OOOOO 


88 


+ ■00774 


06 


+ •00118 


16 


-3 


— •00012 


49 






+ •00027 


29 


— 'OOOOO 


14 


-4 

-s 


— 'OOOOO 


19 






+ 'OOOOO 


73 






Sum 


+ •36714 


54 


-■00532 


72 


+ •02367 


90 


—■02432 


21 



i. 


2» + c + ra. 








2i — c — m. 




1 


5 








— 'OOOOO 


18 




4 


+ 'OOOOO 01 






— -00006 


09 






3 


+ '00003 75 






-•00323 


90 


+ 'OOOOO 


50 


2 


+ '00182 63 


+ 'OOOOO 


37 


-•10552 


73 


+ '00060 


49 


1 


+ -04846 15 


+ '00044 


97 


-•59901 


89 


+ -03886 


04 





+ 1-24970 06 


+ '02492 


64 


+ •37321 


43 


+ -00038 


74 


— 1 


- -30203 78 


+ •00957 


00 


— -11431 


89 


— '00019 


28 


~ 2 


+ -06215 35 


+ -00498 


5i 


— •00316 


98 


— 'OOOOI 


52 


-3 


+ '0=254 52 


+ -00004 


12 


— '00005 


58 


— 'OOOOO 


02 


-4 


+ -00005 27 


+ 'OOOOO 


04 


— 'OOOOO 


05 






-5 
Sum 


+ -ooooo 06 
+ 1-06274 02 












1 
95 


+ •03997 


65 


-■45217 


86 


+ '03964 



Motion of the Moon. 



25 



Characteristic ee'k 2. 
Values of & + 1. 



i. 


2i + c — m. 




28- 


-c+ m.. 




5 


+ -ooooo 


03 






— 'OOOOO 


03 






4 


— 'ooooo 


41 






+ •00000 


95 






3 


— -0002I 


94 


— "OOOOO 


02 


+ •00052 


42 


— 'ooooo 


07 


2 


— -01004 


62 


— '00002 


58 


+ •01909 


27 


— -00008 


59 


1 


— -19834 


06 


— -00318 


89 


+•45975 


74 


— •00548 


S7 





— 1-20621 


23 


-•03223 


59 


-•47842 


15 


— ■01404 


66 


— 1 


+ '16517 


35 


— -0020I 


12 


+ •20668 


08 


+ •00052 


86 


— 2 


— -02140 


03 


-•OOO73 


39 


+ •01059 


82 


+ ■00010 


47 


-3 


— -00061 


57 


— 'OOOOO 


60 


+ •00024 


04 


+ •00000 


13 


—4 


— -ooooi 


18 






+ 'OOOOO 


40 






-5 


1 — -ooooo 


03 














Sum 


' —1-27167 


19 


— •03820 


19 


+ •21848 


54 


-■01898 


73 



Values of 



i. 


+ -OOOOO 2 

+ •00079 3 

-•14433 
+ -04641 
+ -00043 6 
+ -ooooo 4 


(«','*"% 


(«,*'% 


(ey£ 2 )i. 


4 
3 
2 
1 

— 1 

-3 

-4 


+ -ooooo 7 
+ -00047 5 
+ •01895 
-■08133 

— •05729 2 

— '00121 6 

— •ooooi 5 

— '12042 


— •ooooo 5 

— '00020 I 

+ •01155 
-•13724 

+ •15104 I 

+ ■002X6 2 
+ '00002 2 


— -ooooo I 

— 'OOOII 6 

— •OIO09 I 

+ ■04755 

— •18609 

— •00070 8 

— 'ooooo 4 


+ •02733 


-■14946 


Sum 


— -09669 



Royal Asteon. Soc, Vol. LIV. 



26 



Mr. Ernest "VY. Brows, Theory of the 



Characteristic ee'k 2 



Values of 



i i- 


O,'* 2 ),. 


(«V 2 )i. 


("I"*"),. 


(>'#•% 


\ 4 
3 

2 
I 

O 

— I 

' — 2 

—3 

—4 


+ 'OOOOO 2 
+ ■00014 ° 
+ '002l6 2 
+ •03509 

— '00641 

— '00013 7 

— 'OOOOO 1 


— 'OOOOO 2 

— 'OOOI5 I 

— '01142 
+ •03952 

+ •08429 
+ •00694 7 
+ •00009 2 

+ 'OOOOO 1 


+ 'OOOOO I 

+ '00000 8 

— '00028 7 

— •12043 

+ •14853 

— 01646 7 

— '00027 5 

— 'OOOOO 3 


+ •00002 
+ •00146 6 
— •07174 
+ •52873 
+ •00182 4 
+ '00001 3 


Sum 


+ •03085 


+ •11928 


+ •01108 


+ •46031 



i. 


(«?K-')i. 




(*'i)'M:') ( . 




(ei;'M% 


(«W)i- 


4 






— 'OOOOO 


I 






3 


+ 'OOOOO 


2 


— '00015 





— 'ooooo 6 


+ 'ooooi 9 


2 


+ "00009 


7 


— '00960 


3 


— '00039 1 


+ '00082 7 


1 


+ '00671 


7 


- '35943 




-•01857 9 


— "°833° 





+ '46033 




— '82002 




-•65366 


+ 1-19699 


— 1 


+ -59651 




— '02285 


7 


+ •46862 


+ -05037 5 


— 2 


+ "02033 


9 


— '00017 


9 


—•00525 9 


+ '00064 8 


—3 


+ '00017 


6 


— 'OOOOO 


1 


— -00003 9 


+ -ooooo 7 


—4 
Sum 


+ 'OOOOO 


2 






— "OOOOO 1 




+ 1*08422 


— 1*21224 


— '20932 


+ 1-16557 



Motion of the Moon. 



27 



106. Characteristic </ 2 k 2 . 
Values of M + i- 











! 




2i— 2m- 


-2g. 




i. 




2J + 2m + 


2g. 






_____ -_-—-- 






4 








+ -00004 


18 














+ •00224 


16 






3 


— 'OOOOO 


°3 






















+ •07521 


17 


+ 'OOOOO 


14 


2 


+ 'OOOOO 


75 














1 


+ -00043 


°3 


— 'OOOOO 


08 


-•5399 8 


50 


— •00079 
— -00969 


35 





+ •09250 


43 


+ ■00101 


26 


+ •07918 


70 


97 




—•68816 


3 1 


—■09327 


73 


— •00013 


90 


+ -00002 


11 


— 2 


+ •03528 


74 


-•00795 


57 


+ -00005 


74 


+ -ooooo 


01 





+ ■00101 


97 


— -00005 


78 


+ 'OOOOO 


12 






-4 


+ •00001 


95 


— -ooooo 


05 








__ 


Sum 


-•5S889 


47 


— ■10027 


95 


-•38338 


33 


— ■01047 


06 





. 











11- 


- 2m + 2g. 




i. 




2! + 2m- 


"2g- 








, — 


— 













— -ooooo 


02 






4 


+ -ooooo 


20 






+ •0000 1 


86 






3 

2 

1 



— 1 

— 2 


+ -00006 


94 














+ -00203 
+ •12148 

— -12571 

+ •21424 
+ •00727 


74 

35 
86 


+ -ooooo 
+ •00010 

-•00395 


01 

79 
81 


+ •00116 
+ -04889 

-•11524 


3 2 
60 

84 


— 'OOOOO 

+ •00001 


04 

3° 


14 
26 


— -00030 

— '00000 


39 

43 


+ ■12137 

— '00002 
+ '00002 


41 
34 
67 


—•00037 

— ■00042 

— -ooooo 


3i 

17 

17 


-3 

-4 


+ •00014 


40 














+ 'OOOOO 


18 






] + 'OOOOO 


11 







Sum 


+ •21953 


35 


-•00415 


83 


1 +'05620 


77 


— •00078 


39 








28 



Mr. Ernest W. Brown, Theory of the 









Characteristic 


e'W 














Values of &+%. 








i. 


2! + 2m. 




2i — 2m. 






4 


— '00000 


08 




— "OOOOO 


89 






3 


— '00002 


is 




— "00047 


64 






2 


— ■00156 


93 


— •00000 01 


-• OI S35 


04 






I 


— -11505 


II 


— - ooooo 61 


+ -44404 


18 


+ 'OOOOO 


55 


O 


-•17343 


72 


— ■00016 14 


-•13677 


02 


+ •01266 


87 


— I 


+ •52334 


25 


+ •"337 32 


— 'io334 


50 


+ •00245 


57 


— 2 


+ •02473 


90 


+ •00102 26 


— •00042 


17 


+ '00002 


21 


—3 


+ •00058 


62 


+ 'ooooo 90 


+ 'OOOOO 


55 


+ "ooooo 


02 


—4 


+ •00001 


06 


+ •00000 01 


+ 'OOOOO 


01 






Sum 


+ •25859 


84 


+ •11423 73 


+ ■18767 


48 


+ •01515 


22 



*'. 


21 + 2g. 


2i — 2g. 


4 










— ■OOOO I 


75 




3 


— "OOOOO 


51 






— '00I08 


34 




2 


— "00024 


95 






— •O4714 


61 


+ •00000 19 


1 


— '00940 


95 


+ 'ooooo 


12 


+ •29280 


13 


+ '00080 10 





-'03505 


90 


— •00082 


36 


-•O0795 


42 


— '02647 °3 


• — 1 


+ •32065 


65 


+ •03486 


41 


— •O5402 


7i 


— 'oooio 85 


— 2 


— "02496 


30 


+ •00468 


96 


— "00194 


98 


+ 'ooooo 02 


—3 


— '00056 


64 


+ •00002 


20 


— '00004 


05 




—4 


— -ooooo 


9i 


+ 'ooooo 


02 


— "OOOOO 


07 




Sum i 


+ •25039 


49 


+ •03875 


35 


+ •18058 


20 


-■02577 57 



Motion of the Moon 



29 



Characteristic e /2 k : 



'■Ar* 



Values of £ + &. 



4 

3 

2 
1 
o 

— 1 

— 2 
-3 

-4 



Sum 



+ -ooooo 33 

+ •00013 97 

+ -00404 2 1 

— •22402 63 
+ •33808 64 
-•32859 84 

— •01208 10 
— '00024 56 

— •ooooo 33 



— '22268 31 



+ ■0001 1 94 

— •00148 01 

— •05109 26 
—•00033 08 

— 'ooooo 27 



-•05278 68 



Values of 



i. 


(i 2 ^)i. 


(V 2 /fc' 2 )s- 


(V*' 4 )i- 


(v'W\- 


3 

2 
1 



— 1 

— 2 
-3 


+ -ooooo 7 
+ •00167 8 
-•38676 9 
+ •01312 8 
+ -00008 1 


+ -00009 7 
+ •00617 8 
+ •17825 2 
+ •02792 3 
— •00001 8 
+ •00000 1 


+ -ooooo I 

— 'OOOI2 8 

—■07136 

-•08335 9 
+ -02048 7 
+ '00026 7 
+ -ooooo 3 


+ -ooooo 6 
+ ■00033 3 
+ •00475 7 
+ •13082 
•ooooo 

+ 'OOOOO 2 


Sum 


-•37187 5 


+ •21243 3 


— ■13409 


+ •13592 



SO 



Mr. Eenest W. Beown, Theory of the 



Characteristic e /2 k 2 



Values of 



i. 


\vVck% 


W'kk'y,. 


{rm'k 2 ),,. 




{■Wl' k'-)i. 




(WM'h 


3 






— '00002 









— -00004 


6 


+ .00000 68 


2 


— 'OOOO I 


9 


— 'OO 1 42 


2 


— 'OOOOO 


1 


— '00348 


4 


+ '00048 87 


I 


— - OOOOI 


I 


-' 72S3 


2 


+ 'OOOO 1 


5 


-•19279 




+ •02689 8 


O 


+ •27381 




—4IS68 




+ 'OOIOI 





— - oi9i9 


2 


+ •08383 2 


— I 


+ •59306 


6 


-•O5074 


8 


+ •30225 




— '00469 


2 


— •26628 1 


— 2 


+ '00284 


8 


— •OOOO I 


6 


- '00945 


8 


— "00006 


7 


— "OOI2I 51 


-3 


+ 'OOOO 2 


4 


+ 'OOOOO 


1 


— '00004 


3 


— 'OOOOO 


1 


— 'OOOOO 96 


Sum 


+ ■86972 


- '54042 


+ '29377 


— •22027 


— •15628 



107. Characteristic k. 



Values of $. For arguments 2i±4g, 21, 2=o. 



i. 


2i + 4g- 


24 -4g. 


24. 


4 




+ 'ooooo 44 






3 




+ ■00026 86 


+ '00002 


20 


2 


+ 'ooooo 07 


+ ■01037 01 


+ '00183 


54 


1 


+ •00005 65 


— '12623 98 


+ '12460 


70 





+ •00288 65 


+ •02178 99 


-3-96849 


90 


— 1 


— ■09817 71 


+ '00042 43 


+ '09150 


01 


— 2 


+ •01326 43 


+ •00000 58 


+ '00077 


12 


-3 


+ - oooo8 29 




+ 'OOOOO 


57 


-4 
Sum 


+ 'ooooo 24 








-■08188 38 


-•09337 67 


-374975 


76 



Motion of the Moon. 



3i 



Characteristic k*. 
Values of St + &• 



i. 


21 + 2g. 




21 — 2g. 




4 






— -ooooo 03 




! 3 


+ -00000 03 




- -00003 75 




1 2 


+ -00008 09 




+ -00059 77 


— -ooooo 33 


1 

\ I 


+ -OI375 2 5 


+ -ooooo 02 


— -01261 91 


-•00193 31 


! 


+ 1-96250 00 


+ •00025 03 


+ 1-95644 07 


+ -Q39S 1 5 2 


1 
1 —1 


+ -01515 37 


—■00427 50 


- -01125 33 


+ -00008 92 


1 —2 


+ -002II 13 


— •0000 1 36 


— -00040 44 


+ -ooooo 04 


! -3 


+ -00007 78 


— -ooooo 01 


— -ooooo 64 




-4 


1 + "00000 15 

i 








Sum 


+ 1-99367 80 


— ■00403 82 


+ 1-93271 74 


+ •03766 84 









Values of 






i. 


(*% 


+ -oooo 1 3 


J 


(***')(. 


(**"),. 


C^/t"),-. 


3 




— -ooooo 6 


+ •00000 05 


2 




+ •00072 4 






— •00053 8 


+ -00002 68 


1 




— -05111 5 




— •00003 6 


— •03284 


— -OOOI2 O 





+ •00001 1 


+ •00171 4 




— •00096 1 


+ •98818 1 


— •99206 I 


— 1 


+ -00029 6 


+ •00001 5 




+ ■02794 


— •001 2 1 8 


+ •06300 2 


— 2 


+ •00412 






+ •00260 9 


— •0000 1 5 


+ -00004 69 


-3 


+ -00004 2 
+ -00446 9 






+ •00000 9 




+ -ooooo 02 


Sum 


— -04864 9 


+ •02956 


+ '9535 6 


-•92910 5 



32 



Mr. Ernest W. Brown, Theory of the 



108. Characteristic e 8 ct. 



Values of &+&. 



2.1. 




21 + 


3?- 






21- 


3c- 




9 










— -00018 


71 






7 


+ "OOOOO 


19 






— '01 000 


07 


— "OOOOO 


34 


5 


— '0O0O2 


09 






-•09655 


71 


+ 'OOOOO 


30 


3 


— -00939 


13 


— 'OOOOO 


27 


+ -04866 . 


76 


+ '00008 


93 


i 


— -63721 


77 


— -00036 


5 2 


+ •18169 


35 


— '00037 


34 


— i 


+ 1-05819 


96 


+ -00042 


40 


-•05478 


44 


+ •00181 


04 


-3 


+ -05883 


14 


+ -00032 


86 


— '00065 


86 


— '00181 


00 


-5 


— -02176 


80 


+ -00035 


12 


+ 'OOOOO 


19 


— '00003 


19 


-7 


— -00114 


07 


+ -oooo I 


62 






— 'OOOOO 


04 


-9 
Sum 


— "00001 


96 


— - 0000 1 


67 










+ '44747 


47 • 


+ ■00073 


54 


+ •06817 


5i 


— '00031 


64 



2(. 




2.i-i 


- c. 






2J- 


-c. 




9 


+ 'OOOOO 


04 






— '00002 


°5 






7 


— '00005 


87 






— '00220 


26 


— 'OOOOO 


03 


5 


— 00826 


16 


— -ooooo 


17 


— 'I2I22 


64 


— '00004 


70 


3 


- -48587 


63 


— -00023 


37 ' 


— '64460 


95 


— 'OOOO I 


98 


1 


- 71097 


95 


+ ■00184 


02 


+ ■18169 


80 


— •00102 


75 


— 1 


+ '19813 


19 


— '00148 


48 


—•15240 


58 


— ■00236 


47 


-3 


- -13854 


40 


— '00165 


65 


— '04646 


56 


+ '00442 


62 


-5 


— '01271 


97 


+ •00110 


69 


— '00069 


53 


— '00109 


85 


-7 


— '00020 


42 


— 'O0O2 2 


96 


— 'OOOOO 


45 


— 'OOOOI 


93 


-9 


— 'OOOOO 


24 


— 'OOOOO 

— '00066 


40 
32 


— 'OOOOO 

-•78593 


°3 


— 'OOOOO 


03 


Sum 


—1-15851 


41 


2 S 


— '00015 


12 



Motion of the Moon. 



33 



Characteristic e 3 a. 



Values of 



9 

7 

5 

3 
i 

— i 
-3 

-s 

-7 
-9 



Sum 



(«•«)«■ 



— - ooooo 3 
— -00027 4 

— -01911 6 
+ •07964 1 
+ -03460 
—•00230 3 

— -00002 7 

— 'OOOOO I 



+ •09252 



(«"a) f 



— 'ooooo s 

— "00039 8 

—•00925 5 
— -01426 

+ •01030 2 

— '00130 2 

— -00005 9 



— -01498 



(«*«'«),. 



— 'OOOOO 2 

— '00023 9 

— -01591 3 

—•05442 6 
+ •18382 

— -01911 2 

— -00023 8 

— -ooooi o 



+ •09388 






— "ooooo 
— '00006 

—•00436 
-•05569 

— -10225 

— -02624 
— -00089 

— -00004 



-•18955 



109. Characteristic eVa. 



Values of .£ + &. 



2i, 


21 + 2C + m. 


2i — 2C- 


-m. 


2i' + 2C- 


-m. 


9 






— "00009 








7 


+ -00003 




— •00668 




+ 'O0004 




5 


+ -00262 




— •29203 


— -00027 


+ 'OOO36 




3 


+ -16029 




-•2759O 


+ •00137 


- - °3993 


— •00004 


1 


+ 6-03740 


— -ooooi 


-•"737 


— "00649 


- 77599 


— "00262 


— 1 


— +2403 


+ •00261 


+ •74524 


+ "00422 


-1-83937 


+ •00191 


3 


— -12488 


—•04085 


+ •01684 


+ •00599 


+ -15188 


+ •00185 


-5 


- -045 r 5 


+ •00835 


+ •00025 


+ '00009 


+ -03582 


— -00200 


~7 


— -00078 


—•00159 






+ '00071 


+ '00028 


-9 
Sum 


+ -ooooi 


— '00003 






+ '0O002 




+ 5' 6 °55i 


—•03152 


+ •07026 


+ •00491 


— 2+6646 


— '00062 



Royal Astkon. Soc, Vol. LIV. 



34 



Mr. Erxest- W. Brown, Theory of the 



Characteristic e 2 e f a. 



Values of «+ 8. 



2i. 


2J - 2C 


+ m. 


: 2t + m. 


2J- 


-m. 




9 


+ - OOOI2 




+ '00002 




+ *0OOO2 






7 


+ -00688 




+ '00115 




— 'OOO23 






5 


+ •26155 


+ 'OOOOI 


+ '06615 




-•O3842 


— '00002 




3 


+ •39707 


+ '00047 


+ 2-47134 


+ •00001 


— 1-40678 


— '00179 




i 


— '62492 


-■00995 


+ ■89452 


+ "00425 


— ■69566 


+ •00653 




— i 


— •IOO50 


+ '01256 


— •27015 


-'°3575 


+ '3°3° 8 


— •01 102 




-3 


— '00121 


-•01973 


— "20414 


+ -04986 


+ •32001 


—•00456 




-5 


+ 'OOOI2 


— "00042 


— •O0349 


— "01142 


+ •00706 


+ •00259 




-7 

-9 


+ 'OOOOI 


— 'OOOOI 


— '00002 


— "00021 


+ •00010 


+ '00004 




Sum 


-•06088 


-•01707 


+ 2-95538 


+ •00674 


— 1 - 5 1082 


— '00823 











Values of 






2/, 


(e^aV;. 


(«'y«)<. 


( E y°)<. 


(yv);. 


(«'1«)i. 


(«Y°)f. 1 


7 




— '0002 




+ '0003 






5 




-•0177 




+•0129 


+ '0023 


— '0013 1 


3 


+ '005 I 


-•2686 


— 'OOII 


—■0879 


+ '1281 


— 'P7 46 


1 


+ •2742 


-•1039 


-•0435 


+2T242 


+ 1 '446 1 


— •0768 .; 


— 1 


— '0294 


+•0341 


— 1-0316 


— 'OOIO 


-' I 597 


-3-8796 ] 


-3 


+ '4801 


+ '0005 


+ •4427 


— '0009 


— '0009 


+ " OI 35 | 


-5 


— '0003 




+•0013 




— '0007 


+ '0002 1 


-7 


— '0002 




— "OOOI 






' 1 
1 


Sum 


+ 729S 


-•3558 


—•6323 


+ 2-0476 


+ 1+152 


— 4-0186 . 1 



Motion of the Moon. 



35 



no. Characteristic ee' 2 ct. 



Values of j? + i\ 



2i. 



1 9 


2 i + c + 


2m. 








7 


— -00004 


4 




5 


— •00287 


6 




3 


-•13497 


9 


4--ooooo 7 


i 


— 2-50990 


8 


— "00292 


— i 


— •80061 


9 


4- -01047 1 


-3 


-•15688 


6 


+ '34620 1 


-s 


— •00452 


7 


— •05404 8 


-7 


4- - ooo 1 1 


7 


— •00079 1 


-9 






— •0000 1 


Sum 


-3-60972 


2 


4-'2989i 



21 — e — 2m. 



4- 'oooo 1 1 

4- '00026 6 

— ■04866 o —'00005 2 

— •76647 7 — "00666 6 

— •62264 6 4- '03178 7 

— •45296 4 4- '01308 5 
— -01619 4 —'00036 4 

— •00031 5 —'00003 7 

— 'ooooo 2 — 'OOOOO 1 



■ 1-90698 1 4- '03775 2 



24 + C 


-2m. 




4- -oooi 7 






4- -00424 I 


— -ooooo 


2 


— -0220I 2 


—"•00024 


5 


4- '26600 8 


— -01207 


2 


— -47218 9 


+ '01051 


8 


-•06595 8 


4- -00408 


9 


-•00350 9 


+ •00233 


8 



■•00007 7 + -ooooo 5 



-•29332 6 +'00463 1 



2(. 

9 


2i 


— C+ 2m. 








7 


— 'OOo6l 


O 




5 


— •O2829 


2 


— 'OOOOO I 


3 


-'35775 


3 


— •00013 I 


! J 


— -52080 


9 


+ •00478 9 


i - 1 


4- '06614 


8 


4- -09080 3 


! -3 


4- 'O0826 


5 


— •13086 1 


1 -5 


4-'ooo92 


1 


— •00267 4 


! —7 


4- '00003 


6 


— -00003 7 


! -9 






— -ooooo I 



4- -00023 9 
4- -01488 5 

+ •64255 4 

4-2-14351 8 

+ 73440 6 

+ •61312 7 

4- -02246 9 

4- '0004 1 5 

4- 'ooooo 3 



4- 'OOOOO I 

4-'oooio 7 
— '01498 3 

+ '03484 2 
— -11872 9 
4--oi657 8 
4- -00023 1 
4- -ooooo 3 



4- -00004 6 
4- -00360 1 
4- -17806 3 

+ 2'9i3+5 5 
+ •68113 3 

4- "40067 2 
4- -06910 6 
+ •00143 3 
4- -00003 4 



— 'ooooo 3 
4- '00266 3 

+ "01973 I 
4- '02618 8 
4- -04654 8 
4- '00087 ° 
4- '0000 1 2 



Sum 



-•83209 4 --03811 3 



4-4-17161 6 —-08195 o 



-4-24754 3 +-09600 9 



2,6 



Mr. Ernest W. Brown, Theory of the 



Characteristic ee^a. 



Values of 



I 

21. 


(*V 2 a)i. 


(«YM«. 


(«? ,! <0«. 


(«V<0<- 


(«Il'a)j. 


J («W«) 4 . 


5 




— •0023 




— - OOII 


+ '0004 


+ •0071 


3 


— •0050 


— •1649 


+ 'OOOI 


— •0297 


+ •0211 


+ •2706 


i 


-•l86 S 


— '1208 


+ -005 1 


— "1912 


+ •1708 


— •0926 


— i 


— "2071 


— •0297 


— •0445 


+ •1006 


+ "4363 


+ "0461 


-3 


+ •4312 


— '0004 


— '0004 


— -0126 


— •0316 


+ •0067 


-5 


— - oo6i 




+ •0001 


— •0001 


+ "0023 


+ •0001 


Sum 


+ '0265 


-•3181 


— •0396 


— '1341 


+ '5993 


+ •2380 



in. Characteristic e' l a. 









Values of £ + £. 










2i. 




2i + 


3m. 






25- 


3m. 




9 










+ •00001 


4 






7 


+ '00000 


9 






+ •00037 


I 






5 


+ -00043 


4 






+ •00820 


4 


— 'OOOOO 


4 


3 


+ •00474 


3 


+ '00006 4 




-■19909 


9 


— "00062 





1 


—•29874 


9 


— •00700 1 




— •04918 


3 


-•03895 


6 


— 1 


— '00992 


9 


+ "°5°54 9 




— •09107 


7 


+ '00285 





-3 


-■03587 


3 


— •60765 5 




+ •00026 


9 


— "00063 


4 


-s 


+ "00204 


6 


— •00855 2 




+ '00004 


9 


— "OOOOO 


8 


-7 


+ '00006 


6 


— 'OOOIO 




+ 'OOOOO 


1 






-9 


+ "OOOOO 


2 


— -ooooo I 












Sum 


-•33725 


1 


-•57269 6 




-'3304s 


1 


~'°3737 


2 



Motion of the Moon. 



37 



Characteristic e' 3 a. 



Values of Jt+g. 



2%. 


2i + m. 




2i- 


m. 




9 










+ - 00000 


9 






7 


— '00015 


9 






+ '00040 


8 






S 


—•00858 


7 






+ '02460 


2 


+ 'OOOOO 


I 


3 


— •26123 


8 


— '00002 


3 


+ -83332 





+ '00004 


5 


i 


-•01739 


6 


— '00566 


6 


+ -25130 


2 


— •00662 


8 


— i 


+ •08868 


9 


-•03773 


4 


— '00639 


3 


+ •01028 


3 


-3 


+ - i43°S 


5 


+ •31427 


9 


— '04100 


1 


—•00738 


4 


-5 


+ •00288 


4 


+ '00347 


8 


— '00104 


5 


— -00021 





-7 


+ '00004 


4 


+ '00004 





— 'OOOOI 


8 


— 'OOOOO 


4 


-9 


+ '00000 


1 


+ '00000 


1 










Sum 


— •05270 


7 


+•27437 


5 


+ i'o6n8 


4 


— •00389 


7 



Values of 



21. 


(^)i. 


0?"«)i. 


0?Y«)<. 


(rm'*«U- 


5 


+ •00001 


+ -00036 


— '00028 


+ •00072 


3 


— •00012 


+ •00972 


— -01429 


+ •03354 


1 


— •0723 


+ •3042 


-•3287 


-•2531 


— 1 


-•7294 


+ •0898 


+ •7869 


+ •9756 


-3 


-•1709 


— '00002 


+ •08589 


—•00436 


-s 


— -00046 




+ •00028 


— -00004 


Sum 


-•9732 


+ +04I 


+ •5298 


+ •7524 



38 



Mr. Ernest W. Brown, Theory of the 



112. Characteristic elcos. 



Values of jt + g. 



21. 


2i+C + 2g. 




2! 


-e-2g. 


9 










— 'OO002 


63 




7 


— '00000 


06 






— -00232 


93 




5 


+ 'OOOOO 


J 3 






— •12694 


76 


— -00003 62 


3 


— '00042 


98 


+ -ooooo 


01 


+ •18390 


29 


— •00016 14 


i 


- -05518 


61 


+ •00002 


60 


+ •23086 


82 


+ •00590 31 


— i 


+ 1-98177 


62 


— -00800 


76 


—•54487 


76 


—•00573 30 


-3 


+ '21544 


85 


+ •01729 


16 


—•00453 


24 


— '00009 3 1 


-s 


— -03239 


84 


— •00025 


38 


+ '00001 


95 


— "OOOOO 10 


-7 


— -00064 


34 


— -00017 


96 


+ 'OOOOO 


06 




-9 


— -ooooo 


68 


— -ooooo 


19 








Sum 


+ 2-10856 


09 


+ •00887 


48 


— •26392 


20 


— 'OOOI2 l6 



zi. 




2i + e- 


-2g. 






21- 


-C+2g. 




9 


— 'OOOOO 


09 














7 


— '000 1 3 


36 






— 'OOOOO 


25 






5 


— '01084 


86 


— 'OOOOO 


*5 


— '00030 


68 






3 


-•51891 


28 


+ 'OOOI4 


81 


— '01792 


58 


— 'OOOOO 


56 


I 


-•14053 


5i 


— •OO594 


66 


+ 170964 


27 


+ '00054 


14 


— I 


+ '54234 


01 


+ -OI3IO 


49 


- -18777 


01 


+ •02116 


52 


-3 


-•14735 


54 


+ - OII24 


11 


— '11929 


11 


—■02572 


49 


-5 


— '00168 


14 


+ '00004 


72 


— '00276 


70 


+ •06047 


81 


-7 


— 'ooooo 


68 


+ 'OOOOO 


02 


— '00003 


48 


— 'OOOOO 


26 


—9 










— 'OOOOO 


01 


— 'OOOOO 


01 


Sum 


—•27713 


45 


+ •01859 


34 


+ 1-38154 


45 


—•00354 


85 



.Motion of the Moon, 



39 



Characteristic ek 2 a. 



Values of jt + g. 



21. 


2i + C. 




21 - C. 






9 










+ -ooooo 


34 






7 


+ '00001 


07 






+ -00023 


93 


+ -ooooo 


03 


5 


+ - OOI26 


86 


— -ooooo 


10 


+ -02399 


07 


+ -00002 


16 


3 


+ - U95 8 


89 


— -00009 


19 


+ 1-50798 


22 


— -00409 


44 


i 


+ 675069 


59 


+ •01948 


61 


-178455 


48 


— -02I57 


33 


— i 


-1-93792 


53 


— -04226 


85 


+ 1-07322 


87 


— •O2465 


41 


—3 


+ -19026 


33 


— ■02071 


99 


— -02962 


9i 


+ •00703 


99 


-5 


— -00889 


43 


+ •00496 


60 


— -00036 


56 


+ -OOOI3 


23 


-7 


— -OOOI2 


71 


+ -00004 


56 


+ -ooooo 


03 


+ -ooooo 


15 


-9 


— -ooooo 


10 


+ -ooooo 


04 










Sum 


+ 5-11487 


97 


-•03858 


32 


+ -79089 


5 1 


—•04312 


62 



Values of 



2/. 


(«**■).-. 


(«'A' 2 o)i. 


(e£'-a);. 


(e'Fa\-. 


7 






— -OOOI2 


7 


— -ooooo 


5 






5 






— -01181 


1 


— -00052 





+ -ooooo 


4 


3 


+ •0000 2 


3 


— •08718 


3 


— -04041 


2 


+ •00064 


2 


1 


+ •00187 


9 


— •O3664 


1 


— '2022O 




+ •11156 


3 


— 1 


+ •15508 


4 


— -04612 


8 


+ •17870 


T 


+•10583 




-3 


+ •15427 


2 


— •OOOI7 


7 


— •OI258 


5 


—•04372 


5 


-5 


— ■00719 









— '00006 


3 


— -00007 


6 


-7 


— -00005 


1 






— -ooooo 


1 


— -ooooo 


1 


Sum 


+ •30401 


7 


— •18206 


7 


— -07709 




+ ■17424 



40 



Mr. Ernest W. Brown, Theory of the 



Characteristic ek 2 a. 



Values of 



2J. 


(«/M'a)i. 


^kk'a) t . 


7 

5 

3 
i 

— i 
-3 

-5 
-7 


4- "ooooo i 
4- "oooo6 3 
4- '00669 3 
+ -56588 5 
— 2-22530 

— -10800 4 

— -00080 9 

— "ooooo 5 


4- -ooooi s 
4- "00166 1 
4- "16219 6 
+ i"44U3 

— -03941 i 

— -00373 9 

— "ooooi 5 


Sum 


— 1-76148 


4-1-56184 



1 1 3. Characteristic e'k 2 a. 



Values of & 4- g. 



21. 


2i + m + 2g. 




2( — m - 


-2g- 




9 










— -ooooo 


71 






7 


4- "ooooo 


18 






— -00049 


35 






5 


4- "00017 


33 






- -03515 


46 


— -ooooo 


07 


3 


4- -01544 


48 


4- 'ooooo 


02 


— -07602 


70 


— •00097 


56 


1 


4-1-06559 


78 


4- "00041 


62 


— '28776 


46 


-•00955 


78 


— 1 


- -60593 


45 


-•04399 


71 


4-3-05205 


66 


— -01139 


56 


-3 


4- -00478 


°7 


— "ii86o 


46 


+' °5553 


97 


4- -00002 


02 


-5 


- -01154 


32 


— •00649 


08 


4- "00070 


44 


4- -ooooo 


06 


-7 


— -00014 


63 


— •00006 


41 


4- "ooooo 


78 






-9 


— -ooooo 


21 


— •ooooo 


05 










Sum 


+ -46837 


23 


-•16874 


07 


4-2-70886 


17 


— •02190 


89 1 



Motion of the Moon. 



4i 



Characteristic e'k^a. 



Values of M + S. 



2i. 




2i + m- 


-2g. 




2i — m + 2g. 


9 


+ 'OOOOO 


20 














7 


+ '00021 


14 






+ -ooooo 


18 






S 


+ •01621 


97 


— "OOOOO 


34 


+ -00007 


23 






3 


+ •49466 


56 


+ -00069 


13 


+ -00293 


93 


+ 'OOOOO 


05 


1 


—•97149 


67 


-•00578 


5° 


+ -07585 


28 


— -00011 


26 


— 1 


— '02020 


43 


—•04771 


S3 


— 2-92221 


25 


-•01749 


37 


-3 


+ "00345 


46 


— •00060 


99 


+ -18175 


84 


+ •02886 


88 


-5 


+ •00035 


14 


— -ooooo 


62 


+ -00704. 


5o 


+ •00308 


27 


-7 


+ 'OOOCO 


9i 






+ -00009 


40 


+ •00001 


74 


-9 


+ 'OOOOO 


02 






+ 'OOOOO 


06 


+ -ooooo 


01 


Sum 


-•47678 


70 


-•05342 


S5 


—2-65444 


83 


+ •01436 


3 2 



21. 


2i + m. 




2t — m. 






9 


— 'OOOOO 


04 






— -ooooo 


10 






7 


— -ooooi 


65 






+ -00003 


17 






5 


— '00150 


99 






+ -00417 


5 2 


+ -ooooo 


02 


3 


— -09289 


49 


+ •00002 


48 


+ -33260 


II 


+ -00008 


40 


1 


-1-26453 


15 


— •00687 


78 


+ 1-00154 


29 


+ •01019 


64 


— 1 


+ -43988 


60 


+ •19119 


70 


- -44129 


44 


+ ■02820 


24 


~3 


- -08255 


25 


+ •11614 


20 


+ -19227 


95 


—•03913 


82 


-5 


— -00097 


39 


+ •00120 


59 


+ -00317 


00 


— •00025 


30 


-7 


+ -ooooo 


62 


+ '00001 


11 


+ -00003 


92 


— 'OOOOO 


19 


-9 


— 'OOOOO 


02 






+ -ooooo 


04 






1 Sum 


— i - oo258 


76 


+ •30170 


30 


+ 1-09254 


46 


— •00091 


01 



Royal Astron. Soc, Vol. LIV. 



q 



42 



Mr. Ernest W. Brown. Theory of the 



Characteristic e'k 2 a. 



Values of 



2J. 


(>)£-<«);. 


(V'k"a) t . 


(„£'%);. 


(l'A 2 «)<- 


7 

5 

3 
i 

— i 

-3 

~5 
-7 


+ -OC004 4 
+ •00358 8 
+ •10476 

■+'744i7 

— •00328 7 

— 'OOOOI I 


— '00002 9 

— •OO335 8 
-•34996 
-•68238 

+ '50351 5 
+ •00284 3 

+ •00001 9 


+ •00001 I 
+ •00142 I 
+ •17656 

-•02358 
-•01345 I 

4- -00013 7 
+ - ooooo 9 


+ 'OOOOO I 

+ '00008 6 
+ '00346 5 
— "48005 
-•20583 
+ ■00157 1 
+ - ooooo 5 


Sum 


+,•84926 J --52935 


+ ■14111 


-•68075 



2i 


(#'a) f . 


(I'M'aV. 


7 
5 
3 
1 
— 1 
~3 

-s 

-7 


— "OOOOO 2 

— -00026 7 

— '03252 O 

— 4'3 r 577 
— 1-32620 

— -00148 2 

— -ooooi 7 


+ -OOOOO 2 
+ -00022 O 
+ "02120 2 

+ 7I84I 
+ I2"282l6 
+ -03487 2 
+ -OOOI6 4 
+ "OOOOO I 


Sum 


— 5-67626 


+ i3'°57°3 



Motion of the Moon. 



43 



1 1 4. Characteristic e 2 a 



fin* 



Values of jt + g. 



i. 


2t + 2C. 


2i- 


-2C. 


22. 




4 






+ •00028 




+ '00007 




3 


+ - 000l8 




+ ■01627 


— - O0OO4 


+ '00281 




2 


+ •00754 




+ •29412 


-•OOO33 


+ -12784 


— •OOO35 


1 1 


+ •26891 


— •00061 


— •12528 


-•OO745 


+ ro2879 


+ •00473 





-•37623 


+ •03189 


— •06085 


+ - ooo66 


- - 3°835 


- -00958 


— 1 


— T4252 


— •01204 


+ •02427 


+ '00084 


+ ' 2I 573 


— •OI47O 


— 2 


+ •06237 


-•03041 


+ ■00086 


— '00632 


+ '012 20 


+ •02769 


-3 


+ •00174 


+ ■00495 


+ '00003 


— '00008 


+ '00031 


-•OO245 


-4 


+ -00004 


— •00024 






+ 'OOOO I 


— -00003 


Sum 


-•17797 


— •00646 


+ •14970 


—•01272 


+ I-0794I 


+ •00531 



Values of 



i. 


(«•<•'),. 


(/-a-),. 


(«'« 5 )f. 


3 




+ •00069 


+ -OOOOS 


2 


+ •00018 


+ "033° 


+ '00442 


1 


+ •00829 


+ •035 


+ '08709 





— •02731 


— •0025 


-•079S 


— 1 


— "°95 


+ •00077 


+ •0172 


— 2 


— •0249 


— -00020 


+ •00291 


-3 


+ •00054 




— '00008 


Sum 


-•138 


+ •067 


+ •0321 



44 



Mr. Ernest W. Brown, Theory of the 



115. Characteristic eeV. 
Values of ,)? + £. 



i. 


2J + C + D1. 


2i— 


2 — m. 


4 






+ -0002 I 




3 


— '00009 




+ ■00753 


— •0000 1 


2 . 


— '02560 




+ •26215 


— •00263 


1 


-2-75577 


— -00278 


+ •20795 


+ -00899 





+ 1-34099 


— •00605 


+ •41844 


+ •00925 


— 1 


+ -08386 


+ •01829 


-•34740 


+ •01662 


— 2 


+ "045 ro 


+ •24946 


— "00299 


+ '00849 


-3 


+ -00099 


— -02085 


— "00002 


+ '00007 


-4 


+ -00003 


— ■00024 






Sum 


-1-31049 


+ •23783 


+ •54587 


+ •04078 



i. 


2» + — m. 


2i — c + m. 


4 


+ -00003 




— -O0O06 




3 


+ •00125 




- '°0753 




2 


+ '03440 


— -00003 


— '72035 


— •OOO53 


1 


+ •23283 


— -00644 


— I-I360O 


+ - 0053I 





+ •61656 


+ •02227 


+ -22661 


+ '02298 


— i 


-•40173 


+ •00163 


+ '03323 


+ •05310 


— 2 


— -10185 


—•05084 


+ -00479 


-•07805 


-3 


— -ooioo 


+ •00181 


+ -0002I 


— -OOI02 


-4 


— -ooooi 


+ 'OO002 


+ -ooooi 


— 'OOOOI 


Sum 


+ -38048 


-•03158 


-1-59909 


+ -OOI78 



Motion of the Moon. 



45 



Characteristic ee'a 2 . 

Values of 



i. 


(*ya'% 


(Wh («)'<«% 


( e 'iOi- 


3 

2 
I 
O 

— I 

— 2 

-3 


— '00001 

— '00131 

—•12949 
+ ■1225 
—•1899 

+ •0662 

— "00I2I 


+ •00026 

+•00751 
+ •1213 

+ -275 1 
—•01338 

+ •00047 


+ •00003 
+ •00106 
+ •01164 

+ •2975 
— •2103 
-•01635 
+ •00010 


—•00037 

-•03513 

— '1270 

-•5059 
+ •01300 
— '00417 

— '00002 


Sum 


-•1332 


+ '39!3 +'0837 


-•6596 



116. Characteristic k s a 2 . 
Values of J? + S. 



i. 


2! + 2g. 


2X- 


-2g. 


2J. 


3 


+ •00001 




+ •00037 




+ "00004 




2 


+ •00047 




+ -i53 l6 


+ "00009 


- 'O0055 


+ -00002 


1 


+ •01697 


+ •00027 


— -22860 


+ •00117 


— 1-50841 


— -00l80 





-•46493 


-•03704 


—•07083 


— '10214 


+ 74163 


+ •10091 


— 1 


-•23351 


— -03121 


+ •08806 


— -02647 


— "10082 


+ T3475 


— 2 


+ •04289 


-•03335 


+ •00279 


— - ooo 1 8 


+ "00400 


+ - 02l64 


-3 


+ ■00017 


— •00009 


+ •00005 




+ '00012 


+ •00014 


Sum 


-•63793 


— T0142 


— •05500 


—•12753 


- "86399 


+ - 25566 



Values of 



.-. 


(W) 4 . 


(*«««),. 


(K-V);. 


3 




+ •00015 


— 'OOOOI 


2 




+ •02250 


— '00110 


1 


+ •00008 


+ •632 


— -17168 





-■03354 


— •0508 


+ •2109 


— 1 


— •623 


+ •00535 


+•2299 


! — 2 


— -0300 


+ -00009 


+ •00274 


! ~ 3 


— '00002 




+ •00001 


j Sum 


-•686 


+ •609 


+ •2708 



46 



Mr. Ernest W. Brown, Theory of the 



117. Characteristic ke 3 . 
Values of K+L. For arguments 2t'+g±2c, L=o. 



i. 


2J + g+3C. 


2i + g-3C. 


5 




+ •00000 56 


4 


+ •00000 03 


+ •00022 74 


3 


+ "ooooi 44 


+ 'o°S33 39 


2 


+ •00074 35 


+ •00402 52 


1 


+ •03004 59 


— "01541 20 





+ •71696 43 


— - o75°3 40 


— 1 


— - ioo8i 77 


—•02229 53 


— 2 


+ •00022 10 


— '00089 °° 


-3 


— '00021 24 


— '00002 15 


—4 


— -00015 2 8 


— -ooooo 03 


-5 


— '00000 65 




Sum 


+ '64680 00 


— '10406 10 



i. 




2i+g + C. 






2i + g-e. 


s 


+ 'OOOOO 


OI 






+ 'OOOOO 


16 






4 


+ •00000 


83 






+ '00008 


52 






3 


+ -00043 


36 






+ •00340 


21 






2 


+ •01736 


So 


— -OOOOO 


07 


+ •07930 


55 


— "OOOOO 


12 


1 


+ •40822 


H 


— -00007 


16 


+ •00652 


77 


— - OOOII 


17 





-•01854 


4i 


—•00635 


S3 


+ •06525 


69 


+ •00001 


00 


— 1 


+ •00110 


98 


+ •00022 


09 


— '02001 


07 


— 'OOOOI 


61 


— 2 


— '00204 


37 


— '00003 


89 


— 'OI2I I 


33 


— 'OOOOO 


02 


-3 


— '00229 


06 


— 'OOOOO 


04 


— '00050 


44 






— 4 


— '00009 


■ 7 '6 






— 'OOOOI 


26 






— 5 


— 'OOOOO 


25 






— 'OOOOO 


02 






Sum 


+•40415 


97 


— '00624 


60 


+ -I2I93 


78 


— •0001 1 


92 



Motion of the Moon. 



47 



Characteristic ke 3 . 
Values of 



i. 


<*•'),- 


- — 


(lce' 3 Y. 


(*«'«% 


(W) r 


4 




+ 'OOOOO 


70 


+ •00000 


01 


+ -ooooo 


14 


3 


+ "00000 


01 


+ •00038 


82 


+ 'OOOOO 


68 


+ '00009 


92 


2 


+ •00001 


12 


+ •00180 


89 


+ '00048 


'5 


+ •00531 


28 


I 


+ ■00079 


37 


+ '01349 


47 


+ •02534 


49 


+ •00201 


07 


O 


+ ■04139 


88 


— '02230 


57 


— '00708 


02 


-•°SS35 


33 


— I 


—■02439 


3i 


— '00142 


20 


— '00106 


86 


— •00749 


18 


— 2 


+ ■00007 


65 


— "00002 


5o 


— '00094 


57 


— -00083 


02 


-3 


— 'OOOI2 


54 


— 'OOOOO 


°3 


— '00016 


94 


— •00001 


49 


—4 


— 'OOOOI 


23 






— 'OOOOO 


3° 


— 'OOOOO 


02 


Sum 


+ •01774 


95 


— -00805 


42 


+ •01656 


64 


— •05626 


63 



118. Characteristic ke 2 e'. 
Values of K+L. 



i. 

5 




2t+g + 2C + m. 






2i + g — 2C — m. 












+ 'OOOOO 


66 






4 


— 'OOOOO 


04 






+ '00030 


69 






3 


— '00002 


30 






+ •01032 


92 


+ 'OOOOO 


°5 


2 


— '00107 


37 


+ 'OOOOO 


01 


+ •16457 


95 


+ '00004 


19 


1 


-'03569 


68 


+ 00001 


10 


— •01782 


22 


+ •00016 


40 





-•5456o 


36 


+ •00091 


97 


+ •00562 


27 


— "00079 


83 


— 1 


— •05701 


99 


—•00043 


15 


+ '04641 


16 


— '00003 


02 


— 2 


+ •00702 


83 


— 'OOOOI 


58 


+ •00202 


05 


— "OOOOO 


03 


3 


— '00663 


19 


— 'OOOOO 


13 


+ '00005 


09 




t 


— 4 


— '00036 


53 






+ 'OOOOO 


09 






-5 


— 'OOOOI 


01 














Sum 


-•63939 


64 


+•00048 


22 


+ •21150 


66 


— -00062 


24 



48 



Mr. Ernest W. Brown, Theory of the 



Characteristic keV. 



Values of K+L. 



i. 




2t + g+2C- 


-m. 






2t + g- 


-2c + m. 




5 










—•00000 


08 






4 


+ •00000 


26 






— -00004 


42 






3 


+ 'OOOI2 


68 






— "00149 


82 


+ "OOOOO 


05 


2 


+ •00539 


69 


+ "OOOOO 


01 


— •02391 


83 


+ -00004 


19 


I 


+■14580 


39 


+ •00001 


10 


— '00160 


56 


+ -00016 


40 


O 


+•65773 


97 


+ -0009 1 


97 


+ •00971 


18 


— -00079 


83 


— I 


—•00337 


70 


—•00043 


15 


-•08389 


85 


— '00003 


02 


— 2 


+ "00043 


36 


— -ooooi 


58 


— '00626 


45 


— -ooooo 


03 


—3 


+ •00177 


32 


— 'OOOOO 


13 


— - oooi9 


66 






—4 


+ -00007 


97 






— "00000 


40 






-s 


+ - ooooo 


20 














Sum 


+ •80798 


14 


+ '00048 


22 


— '10771 


89 


— "00062 


24 



i. 




2i + g + m. 






2i + g— m. 


5 


— -ooooo 


02 






+ 


■ooooo 


12 






4 


— -ooooo 


91 






+ 


•00005 


78 






3 


—•00043 


65 






+ 


■00260 


OI 






2 


— -01468 


°9 


+ "OOOOO 


47 


+ 


•08054 


40 


+ 'OOOOO 


29 


1 


-•23463 


84 


+ "00040 


37 


+ 


•96859 


73 


+ •00025 


48 





— -02250 


64 


+ ■00011 


90 


— 


•01639 


85 


— 'OOOI2 


32 


— 1 


+ •01512 


91 


— "00065 


5o 


— 


•00262 


58 


+ •00018 


93 


— 2 


—■04967 


40 


— "OOOOI 


82 


+ 


•01797 


53 


— "OOOOI 


06 


-3 


—•00305 


52 


— "OOOOO 


02 


+ 


•00079 


78 


— "OOOOO 


01 


—4 


— '00009 


00 






+ 


■00002 


02 






-5 


— "OOOOO 


17 






+ 


•ooooo 


01 






Sum 


-•30996 


33 


— "00014 


60 


+ 1 


•05156 


95 


+ ■00031 


31 



Motion of the Moon. 



49 



Characteristic ke'Y. 









Values of 










i. 


(£e 2 7))i- 


(^V)i 




(A«Y)i 




(*<='%,)« 




4 






+ 'OOOOO 


69 






— 'OOOOO 


09 


3 


— 'OOOOO 


03 


+ '00047 


35 


+ '00000 


16 


— •00006 


38 


2 


— '00002 


10 


+ ■02356 


09 


+ 'OO01O 


98 


— '00297 


65 


I 


—•00134 


79 


+ -03846 


7 


+ •00579 


68 


+ •01507 


7 


O 


-•05587 


76 


+ •04524 


8 


+•07543 


81 


— •04015 


9 


— I 


— •10870 


Si 


+ •00541 


35 


— - 02o6l 


5 


— '01098 


67 


— 2 


— '00801 


92 


+ '00008 


3° 


— '00037 


8 


— '00027 


16 


— 3 


— •001 1 I 


3 2 


+ 'OOOOO 


1 1 


+ '00025 


36 


— 'OOOOO 


42 


— 4 


— "0000 1 


33 






+ 'OOOOO 


37 


— 'OOOOO 


01 


Sum 


-•17510 


26 


+ '"3 2 5 


4 


+ '06061 


1 


-•03938 


6 



i. 


(fe'7i)i 




(X-eeV) 




4 


— 'OOOOO 


01 


+ 'OOOOO 


07 


3 


— 'OOOOO 


89 


+ '00005 


55 


2 


— '00058 


9 1 


+ '00344 


09 


1 


— •02666 


26 


+ '12360 


04 





— '12700 


5 


+ '08781 


9 


— 1 


— '02647 


80 


+ •00687 


6 


— 2 


—■00730 


21 


+ •00231 


70 


~3 


— •00014 


20 


+ '00003 


48 


—4 
J Sum 


— 'OOOOO 


20 


+ 'OOOOO 


O^ 


— -18819 





+ '22414 


5 



liOYAL ASTROX SOC, VOL. LIV 



If 



5° 



Mr. Eenest W. Brown, Theory of the 



119. Characteristic. kee' z . 
Values of K+L. 



1. 




2! + g+C- 


-2m. 






2i + g— c 


— 2m, 




5 










+ 'OOOOO 


21 






4 


+ 'OOOOO 


02 






+ •00012 


17 






3 


+ 'OOOOO 


72 






+ ■00499 


31 


+ 'OOOOO 


02 


2 


+ '00020 


64 


— 'OOOOO 


OI 


+ •12823 


94 


+ '00002 


61 


1 


+ •00011 


43 


+ 'OOOOO 


86 


+ '78746 


00 


+ ■00234 


71 





— •17808 


97 


+ •00295 


39 


-■17852 


33 


— •O0867 


68 


— 1 


+ 71766 


93 


— -00484 


°3 


+ '01447 


01 


— 'OOOOO 


22 


— 2 


— '05048 


72 -. 


— '00012 


45 


— '00026 


52 


+ 'OOOOO 


02 


-3 


-•00595 


Si 


— 'OOOOO 


10 


— ' 0000 1 


76 






—4 


— '00020 


16 






— 'OOOOO 


04 






-5 


— 'OOOOO 


48 














Sum 


+ •48325 


90 


— '00200 


34 


+ 7S 6 47 


99 


— -00630 


54 



i. 




28 + g + C - 


-2m. 






28 + g- 


-C + 2H1. 




5 


+ 'OOOOO 


02 














4 


+ •00000 


94 






+ 'OOOOO 


01 






3 


+ '00046 


12 






— '00006 


97 






2 


+ •01699 


66 


+ 'OOOOO 


10 


— '00721 


63 


+ 'OOOOO 


16 


i 


+ •33938 


80 


+ •00009 


10 


-•35 J 59 


54 


+ '0006l 


65 





+ '3°9 I 3 


00 


+ •00473 


20 


+ '31108 


88 


— •01333 


33 


— 1 


— •33037 


02 


— '00144 


85 


-•12455 


46 


— '00037 


07 


— 2 


+ •00733 


70 


+ 'OOOOO 


07 


— •01967 


10 


— "OOOOO 


40 


—3 


+ '00010 


3i 


+ 'OOOOO 


01 


— -00075 


20 






—4 


— 'OOOOO 


°5 






— '0000 1 


81 






-5 


— 'OOOOO 


01 






— 'OOOOO 


01 






Sum 


+ •34305 


47 


+ ■00337 


63 


— '19278 


83 


— "01308 


99 



Motion of the Moon. 



Kl 



Characteristic \ee'' 2 . 
Values of K+L. 



'<■ i 

5 




2i + g + 


e. 




— 'OOOOO 


21 + g- 


-c. 


- - ! 










05 




4 


— 'OOOOO 


29 






— '00003 


45 








— ■00014 


54 






— '00140 


70 


— 'OOOOO 


08 


i 2 


— •00554 


54 


+ •00000 


44 


—■03566 


80 


— '00007 


79 


| I 


— -11838 


75 


+ •00042 


44 


— -20612 


34 


— •00712 


07 


O 


— •05326 


65 


+ •03416 


3 2 


-■03830 


00 


— •00675 


3 2 


— I 


— •13926 


03 


— '00502 


70 


+ •10602 


30 


—•00151 


32 


; — 2 


+ •03413 


5 2 


+ •00010 


36 


+ •01041 


88 


— 'OOOOl 


72 


1 

1 -3 


+ •00264 


53 


+ 'OOOOO 


13 


+ -00033 


81 


— 'OOOOO 


01 


-4 


+ '00007 


99 






+ 'OOOOO 


72 






-s 

Sum 


+ 'OOOOO 


16 














-•27974 


60 


+ •02966 


99 


—•16474 


63 


—•01548 


J' 



Values of 



4 

3 

2 
1 
o 

— 1 

— 2 

■■> 

j 

— 4 
Sum 



(*«.*);. 



+ •00000 01 
+ •00000 52 
— -00002 93 



•04371 



-,6 



— -66115 99 

-•03574 62 

— •00052 45 

— 'OOOOO 69 

— -74117 71 



(*«V)i. 



+ 'OOOOO 2 I 

+ •00015 6l 

+ •00971 71 

+ '35 1 73 9i 

+ •15803 06 

+ •00480 33 

— '0000 1 15 

— 'OOOOO o^ 



+ •52443 



(/.■e„' 2 ) ; . 

+ ■00000 OI 

+ ■00000 75 

+ •00049 78 

+ •02306 46 

+ ■11337 20 

+ •28041 72 

+ •00383 22 

+ •00001 26 



+ •42120 40 



(/«V;j 




— 'OOOOO 


1 

24 


— •00052 


32 J 


— '09968 


00 


-■25777 


09 


-•0595 8 


97 


— •00152 


II : 


— '00002 


38 s 


— "OOOOO 


03 



-•41911 14 



52 



Mr. Ernest W. Brown, Theory of the 



Characteristic kee' 2 . 

Values of 



i. 


(hy<i')t. 


(fe'vv'h. 

— 'OOOOO 06 


4 




3 


— 'ooooo 23 


— '00004 14 


2 


— •00015 34 


— •00245 63 


I 


— "00732 98 


-•07397 30 





— •00507 19 


+ •03863 62 


— I 


+ •12496 93 


+•03793 55 


— 2 


+ •01613 76 


+ •00073 98 


3 


+ '00021 08 


+ '00001 02 1 


—4 


+ 'OOOOO 26 


+ '00000 01 1 


Sum 


+ •12876 29 


+ •00085 05 



1 20. Characteristic ke' 3 . 
Values of K+L. 



i. 




2i + g + 3m. 






2i' + g- 


-3m. 




5 










+ 'OOOOO 


°3 






4 










+ 'OOOOI 


Si 






3 


— 


ooooo 


03 






+ '00070 


49 






2 


+ 


ooooo 


03 






+ '02390 


12 


+ 'OOOOO 


36 


1 


— 


00016 


08 


+ '0000 1 


88 


+ ■39028 


72 


+ '00034 


06 1 





— 


01185 


45 


+ '02137 


21 


— "02187 


35 


+ '02201 


69 \ 


— 1 


+ 


39014 


38 


— "00489 


89 


— '00064 


00 


+ '00003 


96 1 


i 2 


— 


02391 


°s 


— '00002 


03 


— '00019 


99 


+ 'OOOOO 


02 


! _ 3 


— 


00120 


16 


— 'OOOOO 


01 


— 'OOOOO 


23 






j —4 


— 


00003 


10 














! S 


— 


ooooo 


05 














I Sum 


+ 


35298 


49 


+ ■01647 


16 


+ •39219 


30 


+ '02240 


09 



Motion of the Moon. 



i>S 



Characteristic ke' 3 



Values of K+L. 



i. 


. 


2! + g + m. 








21 + g-m. 


. 














5 


4 


+ "OOOOO 


°5 






— -ooooo 


64 






j 


+ '00002 


55 






— '00029 


19 






2 


+ •00062 


54 


+ -ooooo 


OS 


— "00972 


22 


— "OOOOO 


04 


I 


+ •00128 


4t 


+ '00008 


89 


—•15175 


42 


— -ooooo 


49 


O 


-•00357 


95 


+ •01132 


25 


+ •00011 


85 


+ •01084 


69 


— I 


-•14427 


67 


— ■00318 


17 


+ -00047 


58 


+ -00029 


32 


— 2 


+ •01680 


23 


— "0O002 


14 


— •00094 


04 


+ -ooooo 


45 


-3 


+ '0007 T 


34 


— 'OOOOO 


01 


— '00006 


78 


+ -ooooo 


01 


-4 


+ •00001 


71 






— -ooooo 


20 






-5 


+ -ooooo 


°3 














Sum 


-•12838 


76 


+ ■00820 


87 


— -16219 


06 


+ •01113 


94 









Values of 










i. '. 


Oj :, ),> 




(/b)' 3 ),:. 


(AW) ■ 


(Aw' 2 );- 


4 






+ 'OOOOO 02 






— -OOOOO 


01 


3 






+ •00001 59 


+ -ooooo 


05 


— 'OOOOO 


63 


2 






+ •00110 53 


+ '00002 


s° 


— "00041 


80 


I 


— '0000 1 


49 


+ •05659 12 


+ •00025 


06 


— "01944 


00 





+ •00482 


66 


— •00172 9 


+ •06378 


I 


—■06330 


2 


— I 


—•53057 


50 


— '00381 2 


+ •30385 


2 3 


+ '00018 





— 2 


-•00513 


58 


— '00002 82 


+ '00300 


92 


— 'OOOI2 


08 


-3 


— '00006 


46 


— 'OOOOO 01 


+ '00003 


56 


— "OOOOO 


29 


—4 


— 'OOOOO 


oS 




+ 'OOOOO 


04 






Sum 


-•53096 


45 


+ •05214 3 


+'37095 


5 


— "08311 






54 



Mr. Ernest W. Brown, Theory of the 



121. Characteristic k 3 e. 
Values of K+L. For arguments 2i + ^g±c, L—o. 



i. 


21 + 3g + c. 


2( + 3g-e. 


5 






4 




I 


3 


+ 'ooooo 04 


— 'ooooo 43 


2 


+ '00001 74 


— '00041 18 


I 


+ •00087 34 


— "03237 9° 


O 


+ •02556 39 


— 1 '65684 02 


— I 


— •27846 00 


— '11611 13 


— 2 


— •00185 73 


+ '00073 33 


3 


— '00022 02 


— - oooor 73 


—4 


— "ooooi 68 


— 'OOOOO 10 


-5 


— - ooooo 06 




Sum 


— •25409 98 


— 1-80503 16 



| i. 
5 


_ . 




4 


— 'OOOOO 


02 


3 


— '0000 2 


16 


2 


— '00 1 00 


69 


1 


— '02309 


88 





+•86505 


°s 


- 1 


-•09633 


83 


— 2 


—•00055 


86 


3 


— '00016 


69 


-4 


— 'OOOOO 


5 2 


— 5 


+ "OOOOO 


or 



26 + g + C. 








2J + g-C. 












— 'OOOOO 


65 








— 'OOOOO 


01 


— '00036 


08 


+ 'OOOOO 


16 




— 'OOOOO 


57 


— '01471 


73 


+ •00015 


30 




—•00055 


81 


- -24570 


64 


+ '01382 


37 




-'°4955 


90 


+ 2-50685 


95 


— '00124 


00 




+ •00172 


26 


+ -10206 


4i 


+ •00199 


63 




— '00030 


34 


+ -00127 


02 


+ •00002 


21 




— 'OOOOO 


33 


+ 'OOOOO 


58 


+ -ooooo 


02 



+ '743 s 5 4i 



— -04870 



+ 2-34940 86 



+ •01475 6 9 



Motion of the Moon. 



55 



(k 3 e);. 



Characteristic k 3 e. 
Values of 



(*•=*'«),. 



(A-V/e'},- 



4 














— 'OOOOO 


01 


3 






— 'OOOOO 


01 


— -ooooo 


03 


— -OOOOI 


II 


2 


+ •00000 


°3 


— -00001 


35 


— -00002 


72 


— '00106 


94 


I 


+ •00002 


27 


— •00233 


85 


— •00128 


45 


— -08994 


42 


O 


+ •00140 


98 


— ■46095 


5 2 


+ ■23505 


59 


— 2-12659 


97 


— I 


— •06570 


41 


+ •10654 


96 


+ ■07945 


50 


+ -02789 


93 


— 2 


+ •00250 


93 


+ •00095 


26 


+ •00007 


65 


+ -00009 


36 


3 


— - OOOI2 


15 


+ •00000 


06 


— 'OOOOI 


17 


+ -ooooo 


02 


— 4 


— 'OOOOO 

— ■06188 


15 
5o 




45 


— 'OOOOO 

+•31326 


02 
35 






Sum 


-•35580 


— 2-18963 


J4 



122. Characteristic kY 
Values of K+L. 



2t + 3g+l 



4 






3 






2 


— -ooooo 


04 


1 


+ •00020 


12 





+ •02002 


74 


1 


— '10032 


17 


2 


+•00835 


45 


3 


• — '0002 2 


44 


4 


— 'OOOOO 


92 



bum 



— '07197 26 



•00068 29 



2( + 3g-m. 



+ •04217 86 



• — -00068 







+ '00005 


63 






ooooo 


01 


+ •00212 


35 


+ 'OOOOO 


01 


0000T 


65 


+ •02223 


28 


+ '00001 


65 


0006S 


38 


+ •02116 


77 


— •00068 


38 


OOOOI 


56 


—•00348 


!5 


— 'OOOOI 


56 


•00000 


01 


+ •00007 
+ '00000 


67 


■ — 'OOOOO 


1 



56 



Mr. Ernest W. Brown, Theory of the 



Characteristic kY. 



Values of K+L. 



«'. 




2i + g + m. 






2i + g' 


-m. 




4 


+ '00000 


04 






— 'OOOOO 


19 






3 


+ "OOOOI 


83 






— 'OOOIO 


2 5 






2 


+ "00090 


10 


— '00000 


oS 


—•00457 


80 


+ "OOOOO 


40 


I 


+ •02712 


OS 


— "00007 


43 


-•10787 


] 1 


+ "00030 


28 


O 


+ •04871 


79 


— '00030 


12 


+ •04461 


77 


+ "00031 


18 


— I 


— "01092 


69 


+ •00136 


02 


+ •00213 


57 


— "00077 


72 


— 2 


— -00329 


91 


+ •00001 


45 


+ 'OOT08 


00 


— "OOOOO 


49 


-3 


— "OOOII 


59 


+ •00000 


02 


+ •00002 


95 






—4 


— 'OOOOO, 


26 






+ "OOOOO 


08 






Sum 


+ •06241 


36 


+ '00099 


86 


—•06468 


98 


— "00016 


35 



Values of 



i. 


(i 3 l)i. 


(/■■V;,:. 


(F/t'7,),-. 


{VJc'v')i 




4 
3 










+ 'OOOOO 


04 


— 'OOOOO 


22 


2 






+ 'OOOOO 


II 


+ '00003 


83 


— '00020 


05 


1 


+ "OOOOO 


74 


+ "00008 


48 


+ '00343 


98 


— '01413 


28 





+ •00180 


29 


+ '00264 


45 


+ •26780 


3 


— "26499 


7 


— 1 


— "16631 


44 


+ •10195 


2 


+ •01283 


48 


+ •01063 


3 


— 2 


—•0084s 


96 


+ -00444 


82 


- -00045 


07 


+ •00015 


08 


-3 


— -00005 


95 


+ '00002 


14 


— "OCOOO 


54 


+ "OOOOO 


14 


-4 


— 'OOOOO 


°5 


+ '00000 


01 


— 'OOOOO 


01 






Sum 


—•17302 


37 


+ ■10915 


2 


+ •28366 





— ■26854 


7 



Motion of the Moon. 



5/ 



123. Characteristic ke 2 a. 



Values of K+L. 



21. 




2 


9 






7 


+ 'OOOOO 


10 


S 


— - oooo5 


78 


3 


— •00924 


98 


1 


—•50561 


36 


— 1 


+•33265 


is 



2! + g+2C. 



5 
-7 
-9 

Sum 



— -11496 98 

+ •00857 86 

+ •00130 57 

+ •00003 01 



•28732 41 







— 'OOOOO 


95 










—•00075 


30 


+ ■00000 


T4 


+ -ooooo 


03 


—•03408 


84 


+ •00010 


98 


+ •00003 


19 


— -14174 


34 


— '00052 


08 


+ •00244 


75 


+•23592 


66 


+ •00565 


45 


—•00237 


97 


+ •08570 


37 


— •00163 


11 


+ •00092 


44 


+ ■02136 


26 


— -00008 


48 


— -00003 


°5 


+ '00044 


80 


— '00000 


10 


— -ooooo 


38 


+ •00000 


48 






+ •00099 


01 


+•16685 


14 


+ ■00352 


80 



— 9 
Sum 



— -ooooo 02 

— -00005 82 

— •00553 48 

— -26471 64 

— •30003 95 

— •26242 77 
+ •05329 6t 
+ •01064 96 
+ •00023 8i 
+ •00000 29 



-•76859 OT 



+ •00000 01 

+ •00001 32 

+ ■00101 68 

— •00456 62 
+ ■00689 56 

— -0002I 48 



40 



- — 'OOOOO Os 



+ •00311 



Royal Astrox. Soc, Vol. LIT. 



Mr. Ernest W. Brown, Theory of the 



Characteristic ke 2 a. 
Values of 



2/. 


(ke- a )i. 




(AVV>*. 


(ka'a)i. 




9 






— 'OOOOO 02 






7 






— '00002 31 


— 'OOOOO 


10 


S 


— 'OOOOO 


10 


— '00241 69 


— '00016 


04 


3 


— '00025 


49 


-•05586 70 


— •01707 


71 


i 


— -03006 


16 


— •20534 29 


— '09528 


85 


— i 


+ •08743 


57 


+ •02665 51 


+ •21904 


30 


-3 


+ •10025 


96 


+ 'OOI4I 20 


+ •02244 


01 


-s 


+ •00501 


11 


+ •00001 32 


+ •00077 


23 


-7 


+ •00010 


56 


+ "OOOOO 01 


+ 'OOOOO 


74 


-9 


+ 'OOOOO 


10 




+ "OOOOO 


01 


Sum 


+ •16249 


55 


-•23556 97 


+ •12973 


59 



124. Characteristic kerf a. 

Values of K+L. 



21. 




2i + g- 


r e + m. 






2J- 


.-g — c — m. 




9 










+ "OOOOO 


4 






7 


+ '00003 









— "00006 


1 






5 


+ -00213 


9 






— "01062 


8 


+ "00007 


6 


3 


+ '11381 


3 


— 'OOOOO 


3 


— '40554 


7 


+ •00611 





1 


+ 3-68451 


4 


+ •00250 


2 


+ ■03619 


5 


— •04405 


2 


— 1 


— '01515 


8 


— -02278 


5 


+ •00627 


8 


— '00129 


8 


-3 


- -16887 


4 


— '00097 





— -12646 


4 


+ •00011 


1 


— 5 


+ -02886 


5 


— '00030 


8 


— •00363 


7 


+ 'OOOOO 


2 


— 7 


+ -00074 


8 


— "OOOOO 


3 


— -00006 


7 






— 9 


+ -ooooi 

















Sum 


+3-64608 


7 


— "02156 


7 


-•50392 


7 


—•03905 


1 



Motion of the Moon. 



59 



bum 



+ - oooo i 9 

-f -00038 6 

— ■01159 3 

—•29154 S 

+ •01202 9 

+ •03803 6 

— '02610 6 

— '00076 4 

— 'OOOOT 4 

— •27955 2 



Characteristic ket-'a . 



Values of K+ L. 



+ •00000 


2 


+ •00028 


s 


+ ■01677 


6 


—•01348 


j 


+ •00033 


8 


+ •00002 


9 



+ •00394 8 



+ 


•00000 


7 


+ 


'00044 


1. 


+ 


'02328 


2 


+ 


'74244 


1 


+ 


•00857 


• 


_ 


•06503 


7 


+ 


'07677 


2 


+ 


'00203 


5 


+ 


•0000 1 


8 



+ 78853 I 



— 'OOOOO 2 
+ -OO043 3 

— '02096 I 
-■OI387 3 



•03544 8 



Values of 



zi. 


(7«?)a):. 




7 






5 


+ •00004 


4 


3 


+ •00452 


5 


1 


+ ■39617 


9 


— 1 


-•08295 




— 3 


+•28615 


6 


~ '5 


+ •00498 


4 


— 7 


+ •00003 


9 



hum 



- '60898 



(*«y<o,\ 



(Z'€Tj f a) ( 



'0062 



(/te'7ja);. 











+ 'OOOOO 


9 


•0004 8 


s 


+ •00000 


8 


+ '0009S 


2 


'°5375 


8 


— '00047 


6 


+•08783 


6 


•°33 1 7 




-•03291 


9 


— -4212 




•i33 2 




+ •1682 




+ •23485 




•oi555 


7 


— '12480 




+ •01082 


7 


•00015 


7 


— ■00374 


8 


+ '00009 


6 


-'OOOOO 


2 


— '00003 


6 







•0866 



6o 



Mr. Ernest W. Brown, Theory of the 



12=;. Characteristic ~ke' 2 a. 



Values of K+L. 



21. 




21 +g 


r 2m. 


2i + g- 


-2m. 






2i + g. 




9 






+ '00000 3 






+ 'OOOOO 


I 




: 


7 


— 'OOOOI 


6 




+ •00007 3 






+ '00009 


3 






5 


— '00094 


7 




+ ■00215 8 


+ 'O000I 


2 


+ •00520 


9 


— '00000 


1 


3 


— '°3544 


6 


+ '00004 7 


+ •02615 5 


+ '00103 


4 


+ •18452 


3 


+ '00004 


4 


i 


, --28833 


7 


+ •01564 3 


+ •20918 6 


+ •06701 


3 


-•10637 


3 


+ •02244 


8 


— i 


+ •14056 


4 


+ 'o6S4S 3 


— '22082 6 


— "00171 


1 


— •09188 


9 


+ ■00184 


8 


-3 


+ ■11342 


9 


-■00751 3 


+ •06191 8 


+ '00002 


3 


— '16626 


7 


+ •00094 


4 


-5 


+ •00284 


1 


— '00005 ^ 


+ •00239 3 


— 'OOOOO 


1 


— '00906 


2 


+ ■00001 


1 


-7 


— 'OOOOO 


3 




+ '00004 8 






— '00019 


9 






-9 














— 'OOOOO 


4 




1 


Sum 


— ■06791 


S 


+ - o7357 4 


+ •08110 8 


+ ■06637 


: 


-•18396 


8 


+ •02529 


4 I 



Values of 



2i. 


(/.„»«), 


(l-v' 2 a) L 


1 
1 

5 

1 

1 3 
i 1 

1 —1 

1 

1 -5 
-7 


— '00002 6 

— •00215 4 

— '07106 2 

— •18423 6 
+ ■05465 
+ •00026 7 


+ 'OOOOO I 

+ '00006 5 
+ •00201 2 
+ •11048 3 

+ •18843 7 
+ •02052 8 
+ •00017 3 

+ 'OOOOO 2 


Sum 


— '20256 i 


+ •32170 I 



(£f1'«)i. 



+ '00000 2 

+ ■00014 9 

+ ■01187 9 

— '02612 1 
+ •07789 9 

— ■06604 4 
—■00069 8 

— 'ooooo 6 

— '00294 o 



Motion of the Moon, 



61 



126. Characteristic Fa. 
Values of E+ L. 



9 






7 


— -ooooo 


01 


5 


4- - ooooo 


65 


3 


4- '00040 


11 


i 


4- '01996 


76 


— 1 


4-74206 


40 


-3 


-•i8S34 


79 


-5 


4- '00258 


76 


-7 


4-'oooio 


36 


-9 


4- 'ooooo 


16 


Sum 


+ •57978 


40 



4- -ooooo 05 

4- -00005 98 

4- -00810 84 

— '00058 52 

— '00003 09 

— 'OOOOO 02 

+•00755 24 



2i + g. 



4- 'OOOOO 01 

4- -ooooo 88 

4- -00077 21 

+ -05639 87 

4-2-48539 91 

4-1-66872 82 

+ -12295 52 

4- -00276 64 

4- -00004 10 

4- -ooooo 07 



■•ooooo 07 

• •00008 47 

-•01078 67 

•-0I733 5 1 

-•00038 94 

-•ooooo 23 



+4-33707 03 



-•02859 89 



Values of 



2i. 


(^)i. 




1 7 ; 






! 5 


4- -ooooo 


01 


1 3 


4-'ooooi 


17 


! 1 


4-'ooi32 


75 


— 1 


+ -i9 2 43 


08 





4- -16488 


20 


-5 


4- -00249 


9 1 


_ 7 


4- -oooo 1 


21 



Sum 



+ •36116 33 





(Wo),-. 






+ 


•ooooo 


02 




+ 


'OOOO 2 


57 




+ 


•OO424 


96 




+ 


•77503 


01 




— 


[■41930 


03 




+ 


•O4209 


68 




+ 


■00022 


04 




+ 


•ooooo 


14 




— 


•59767 


61 





62 



Mr. Ernest W . Ukoavn, Theory of the 



127. ( lieaxteierktie kea 2 . 
Values of I{+ L. 



4 


+ '00000 


I 


3 


+ '00013 


8 


2 


+ •00567 


* 


I 


+ ■14568 


3 


O 


"•03785 


2 


— I 


+ •20850 


9 


— 2 


-•°355 2 





-3 


— '00131 


5 


—4 


— '00002 


9 


Sum 


+ '28528 


7 



+ '00007 6 

+ '00810 o 

+ •00671 5 

— •02037 7 
+ ■00131 5 

— '00004 8 



— '00421 9 



+ '00002 5 

+ '00105 4 

+ •03302 6 

+ •27137 4 

— ■04866 7 
-■07390 1 

— '00479 ° 

— '00014 ° 

— 'ooooo 4 

+ •17797 7 



+ •00001 5 

+ •00152 3 

— •03529 1 

— '02091 6 

— '00240 4 

— '00026 =; 



-■05734 1 



Values of 



(*«•').- 



+ 


■ooooo 


2 


+ 


•00016 


O 


+ 


•00954 


3 


— 


■00952 


1 


— 


•16281 


9 


— 


■01633 


8 


— 


■OOOI I 





-■17909 o 



+ •00003 - 

+ •00238 4 

+ •08233 9 

+ •05848 6 

— '02730 .1 

— '00036 7 

— 'ooooo 4 
+ •11556 9 



Motion of the 'Moon 



128. Characteristic )s.e'a?. 
Values of K+L. 





i. 
4 




2i + g + m. 






4- -ooooi 


2f-rg — 111. 






— -ooooo 


I 






3 








— -00008 


4 






+ •00053 


9 


4- -ooooo 4 




2 


-'01155 





4- -00005 


4 


+ •01512 


i 


4- -00048 8 




1 


-•84138 


2 


4- -01252 


8 


4- -04220 


3 


4- -05 264 







—•00439 


4 


4- -031 15 


9 


— -00709 


1 


+ ■01538 3 




i — 1 


+ •02933 


s 


+ •05394 


8 


— •02860 


8 


-•00535 1 


1 — 2 


4- -00117 


6 


— -00267 


6 


+ •03479 


3 


4- 'OOOOO I 


-3 


— •00031 


2 


— '00002 





4- -00058 







', ~~ 4 
l Sum 


— 'OOOOI 


» 






4- -ooooo 


5 




— '82722 


4 


+ •09499 


3 


+ •05755 


5 


+ •06316 5 






" 















Values of 



i. 


(.«?«'.);. 




1 



— 'OOOOO 


2 


2 


— '00049 


6 


I 


-•09352 


2 


O 


4--I2870 




— I 


-•17592 


4 


— 2 


— •00054 





-3 


— -ooooi 


6 


Sum 


— -14180 





(A„'a-) r 

4--ooooi 2 

4- -00066 o 

4-'oiio8 7 
— •06371 
+ •18589 

4- '00476 3 

4--oooo2 8 

+ •-13963 



Haverford College, L'a., U.S.A. 
1900 May 14. 



(To be continued.) 



■■:.* 



Theory of the Motion of the Moon ; containing a New Calculation of the Expressions 
for the Coordinates of the Moon in terms of the Time. By Ernest W. Brown, 
M.A., Sc.D., F.R.S. 

[Received 1905 January 21 ; read 1905 March 10.] 



PART IV. CHAPTERS VII. -IX. 

The previous parts of this memoir have been published in the Memoirs of the Royal 
Astronomical Society under the same title in 1897, 1899, 1900. The solution of the 
problem undertaken — the motion of the Moon as disturbed by the Sun supposed to 
move in a fixed elliptic orbit — is completed in the present part. 

It was stated in the introduction to Part I. that the main object in view was a 
new and accurate calculation of every coefficient in longitude, latitude, and parallax 
which is as great as one-hundredth of a second of arc, the result not to be in error 
by more than this amount. So far as I am able to see this plan has been carried out. 
A careful examination of the magnitude of the coefficients which would arise with 
characteristics higher than those calculated here, and a comparison with the results of 
Delaunay and Hansen, seem to show that no characteristics which would jriv 



,'e 



coefficients so great as o //- oi have been omitted. There are possibly four or five terms 
whose characteristics are of the sixth order, which approach o'^oi quite closely. 
These omissions are, however, quite unimportant from a practical point of view. In a 
comparison between theory and observation a few such terms produce nothing sensible 
m the differences : it is only when the number of them is great that any effect is 
shown. 

The coefficients in longitude and latitude in every characteristic calculated have 
actually been found to o"-ooi, and in parallax to o"x>ooi. In the former case a large 
number of coefficients between o" - oi and o //- ooi are included; and, in fact, it is not 
difficult to see that there are comparatively few coefficients lying between o //- oo2 and 
o"-oi which are not included in the tables at the end of Chapter IX. Thus the 
theory, for purposes of comparison with observation, is considerably more accurate 
than was contemplated in the original plan. Moreover, a similar remark applies to 
the coefficients in parallax lying between o /; -ooo2 and o"*ooi ; thus the parallax of 

Royal Astron. Soc, Vol. LYII. i 



mini 



S2 



Mr. Ernest W. Brown, Theory of the 



the Moon can certainly be found theoretically from the new table within o"'oi, so far 
as the solar perturbations are concerned. 

To complete the whole problem of the lunar motion, inequalities arising from 
other sources have still to be considered. These consist of the very minute terms 
arising from the parts of the solar disturbing function which are noted in Chap. I. § 4, 
the terms arising from the figure of the Earth, and perhaps from that of the Moon, 
the indirect planetary inequalities and the direct planetary inequalities. The kst is 
the only set which presents serious difficulties at the present time, and an investigation 
of them has already been started. Hill's work * on the inequalities produced by the 
figure of the Earth probably needs but little supplementing, while a new method f for 
investigating indirect planetary inequalities should render the task of calculating their 
coefficients comparatively easy. 

An important question which cannot be left aside is that of the accuracy of the 
computations by which the results have been obtained. It appears unlikely that the 
problem will be again completely solved in the near future, and some assurance is 
needed that the new coefficients, especially where they differ from those of Hansen or 
Delatjnay, are the correct ones. Fortunately the three theories are so entirely inde- 
pendent in their methods that agreement between them all amounts to practical 
certainty. The differences between Hansen and Delaltnay have given rise to much 
discussion in the past. In general my theory confirms the results of Delaunay in 
these cases ; the coefficients in which all three theories differ are those which are 
difficult to determine owing to the presence of very small divisors. The older 
methods, which approximate along powers of m, are theoretically less likely to be 
correct than a method which approximates along powers of the other parameters where 
the convergence is quite rapid. This, however, is a rather different question than that 
of the actual accuracy of the numerical work. For the latter very numerous tests 
have been used, covering almost every detail, as well as large masses of calculations. 
These tests are discussed below in Chap. VII. Sect, (iv), as well as in other papers to 
which reference will there be found. 

To return to the special part of the work now published. For the terms of the 
fifth order in u the homogeneous equations were used for the first time. In spite of 
the fact that this was a change of method involving much extra work the compu- 
tations were thus kept within reasonable bounds ; the expansion of Kii/r s would have 
nearly doubled the actual work done. The non-homogeneous equation was still used 
for z, as most of the multiplications had been obtained in calculating the fourth order 
terms. But for the sixth order in both u, z the non- homogeneous equations were 
used,, and the work was much less than had been expected. 

The final steps consisted in the transformation to polar coordinates, the change of 

* Washington Astronomical Papers, vol. iii. pt. 2, 1891. 
t E. W. Brown, Trans. Amer. Math. Soc, vol. vi. (1905). 



Motion of the Moon. 53 

the arbitrary constants to the Delaunay system, and the insertion of their numerical 
values to reduce the results to seconds of arc. For the first of these the formulae 

\ were found to permit of such arrangements that much previous work could be usefully 

I utilised. In the final reduction to numbers, values for the constants were used which 

I were neither those of Hansen nor those of Delaunay. A selection was made from 

\ modern determinations which I believe will be found to be very close to the more 

I accurate values to be found when a thorough comparison of the completed theory with 

I the observations has been undertaken. 

I Owing mainly to the complicated character of the work which is embodied in the 

1 results below I have been obliged to do much more actual calculation myself than 

% heretofore. All computations which could with advantage be turned over to a 

I computer have again been done by Mr. Ira I. Sterner, A.M.* His speed and 

I accuracy have been fully maintained, and have contributed in no small degree to an 

I earlier conclusion of the work than I had hoped. He has in all spent some three 

I thousand hours on these calculations, extended over seven and a half years ; my own 

I share I estimate at five or six thousand hours since the work was begun on a complete 

I plan in 1895. 

I The following is the table of contents for the whole memoir : — 

I Chapter I. — General Development of the Theory. 

I Section (i). An investigation of the disturbing function used, with the necessary 

I corrections. 

I Section (ii). The two forms of the equations of motion. 

I Section (iii). Development of the disturbing function according to powers of 

J 1 /a', z, e'. 

1 Section (iv). The form of the solution. The general system of notation adopted 

I to represent the coefficients, arguments, &c. 

I Section (v). Method of solution. Preparation of the equations of motion. 

I Section (vi). Exact definitions of the arbitrary constants used in the theory. 

I Section (vii). Methods used for the solution of the equations of condition satisfied 

I by the coefficients. The long and short period terms which give rise to small divisors. 

1 Manner of obtaining the higher parts of the motions of the perigee and node, 

I Section (viii). Details concerning the numerical calculations and the methods used 

i to verify them. 

I Section (ix). Transformation to polar coordinates. 

I Chapter II. — Terms of zero order. Numerical results. 

I Chapter III. — Numerical results for terms of the first order. 

I Chapter IV. — Numerical results for terms of the second order. 

% Chapter V. — Terms of the third order. 

I§ 

p * The expense has been met by grants from the Government Grant Committee of the Royal Society. 



54 Mr. Ernest W. Brown, Theory of the 

Section (i). A brief outline of the application of the general method to terms of 
the third order in the calculation of the series A. 

Section (ii). New method for solving the linear equations when the series A have 
been obtained. 

Section (hi). Modification of the method in order to avoid, as far as possible, the 
loss of accuracy arising with long-period terms. 

Section (iv). The method of calculating the new parts of the motions of the 
perigee and node, and the coefficients arising therewith. Numerical values of certain 
quantities. 

Section (v). The final numerical results for the series A and for the coefficients 
of all terms of the third order in u, z. 
Chapter VI. — Terms of the fourth order. 

Section (i). Formulas and methods of procedure. 

Section (ii). Values of %, u^/sX; A, izjak. 
Chapter VII. — Terms of the fifth order. 

Section (i). Preparation of the equations for u, s. 

Section (ii). The new parts of c. Terms with small divisors. 

Section (iii). The equation for z. 

Section (iv). Nature of the computations. Tests for accuracy. 

Section (v). Values of A, B, w A £ -1 /ax ; A, izjak. 
Chapter VIII. — Terms of the sixth order. 

Section (i). Formulae and methods of procedure for u. 

Section (ii). The homogeneous equation for z. 

Section (iii). Values of A, B, w^ _1 /aA ; iz/&\. 
Chapter IX.— Results in polar coordinates. 

Section (i). Formula? for transformation. 

Section (ii). Change of the arbitrary constants. 

Section (iii). Numerical values of the constants. 

Section (iv). Numerical values of the parts of the arguments and coefficients 
arising from the various characteristics. 

Section (v). The final values of the coefficients in longitude, latitude, and parallax. 
Errata will be found at the ends of Chapters V., IN. 



Motion of the Moon. 



55 



CHAPTER VII. 



Terms or the Fifth Order. 



Section (i). Preparation of the Equations for u. s. 

129. The Homogeneous Equations for u, s. As stated above, it was found necessary, 
in order to keep the calculations within reasonable limits, to change the method from 
the non-homogeneous to the homogeneous form of the equations, as far as u r „ s 5 were 
concerned. 

Equations (6), (7) of Chap. I. are those to be used. They may be written — • 

o = <J>EEf+^m 2 L' + ff 
= D-(us + z-)-Du . ])s—(Dzy-- 2m(iiDs-sDu) + }m 2 (u + s) 2 - 5 mV + 3 u 2 + 4u 3 -D- 1 (D'w 2 +I)'co a ) (i\ 



o = •* = f + ijmlP-' A' +1)- > S(M' ) 



= nDs-sDu~2WMs + D- l 'im i (u i -^)-^-^ + ',^"J r ^^ 



cm <)u 



(2) 



where the constants of integration are omitted, since they contain only characteristics 
of even order, and q takes only the values 2, 3, since characteristics containing a 2 are 
neglected. The new symbols must be defined ; in all cases, unless stated otherwise, 
suffixes represent the orders of the characteristics present in the functions to which 
they are attached. 

The majority of the indices of £ contain c, g. Xow, to the order considered here, 

C = C + C 2 + C 4 , g = g + g 2 +g.„ 

and therefore the operations J), I) 2 , D~ l introduce parts of c, g other than c,„ g ; these 
parts must be separated. Put 

f = D\us + ~J)-Du . Ds-(Dz) 2 -2m(uDs-sDu) + %m i (ii + s) i -3mh\ 
F = uDs—sI)u—2mus+%-m i D~ i (ii?—s' i ), 

L'= 4 - {3co., + 4w ,-i)-i(Z<' w , + i)' w ,)}, 
9m- " ' 

3m 2 ( ' 'dn du os os j ' 

where c a , g are substituted for c, g in the coefficients when the operations D, I> 2 , D~ l 
are performed. Hence 

Sf, S(Df') are respectively the parts due toe — c , g — g in 

D ll {vs + ^)-J)v, . Ds-(Dz) 2 —2m(u.Ds-sDii)—D- l {D'to. 2 + D'ui i ), 
D(v Ds—sDh — 2mns). 

Denote the unknown coefficients of £ ± (- ,+Tl in it^/n by A.,. X'_ .,-. The equations 
$ = o, F=o are linear with respect to all these coefficients, since terms of order higher 



:fi 



Mr. Ekxf.st W. Bkowx. Theory of the 



than 5 are neglected and those of lower order have been found. It is necessary to put 
the equationslnto forms convenient for calculation; the method is implicitly contained 
in § 33. It will be seen from that section that if we equate to zero the coefficients of 

f i+T in 

2(1 +m) 



I [<t> + (1 + 2 m)¥]=o, * 

D^—i — 2m-Hnr ±> 






(3) 



the terms of principal importance, involving the unknowns in the left-hand members, 
are a(X, + X'_ ( ) ; -a{\-\'^), respectively. This fact serves as a guide for the arrangement 
of the equations which now follows. 
130. Let 

9m = D{uf l ) = T 1 (Du-u), Ss = Z>(sif) = £(Ds+s), 

P = D 2 — 1 — 2m + y, 



(21 + 7)-— 1 — •?»' + 



so that at*,), ^'o arc divisible by m 2 and 
then it will be found that 



IrnH 



+|mV+s2 + L')+.V(i + 2 m)D-i| M 2 -s 2 + A' + ^ 2 S( J Df')| (4) 



^ = w S s _ s5u _ 2 (i+m)«a+^!i>- 1 |« 2 -^ + A'+^(^')} 



... (5) 



It is to be noticed that all terms except those in a> 3 are homogeneous products of 
the second order with respect to u, s, and their derivatives. 

There are terms whose characteristics are of orders o, 1 , 2, 3, 4, 5 in "» *. -■ Let 

u a) s a , s a (a = 5, 4. 3) and «!» s *<> s * ( 6 — S-«) 

distinguish the orders, it being noted that c =o and therefore that z, is not present. 

Then 



2 S /n*/\1 



and we easily obtain from equation (5), 

2>-i* = -£<**-«*> +*-' [{IC^-^^-^^+^+f^- 1 i^ + A' + ^f)} (6) 

Eauations (0 are immediately derivable from (4), (6). 

1. 1 Form* for Computation.-^ the following formula* the bar over any expres- 
.ion, at usual, means that i/< has been put for £. The sign 2, denoting summation 
for values of a, is omitted for the sake of brevity when no misunderstanding of the 
meaning can arise. 



awL 



Motion of the Moon. 



57 



mi. 
ill 



i 
II 



m 

P 

mm 

P 
P 

1 



1 

ill 



Put 

G = M a % + M <( % + -W 

H = M a M 6 _7{^r 6 4..i.A' + -^/(-Of) ... 

.A. 



-7> 

(8) 



A=- 



.-'/ . ■--«■ — .-/( 



-zDz a . Dz b + 2z A + (2m-im*)2z A + M+'im*-G + i ) m%- S + 2m)D~m (9) 
B = u a $s h +u a &s„-f 2(1 +m)|-F -1 A+ 2s (1 ^ + m. 2s B S;, + 3in 2 (i + 2m)Z) _1 H-^2(i + 2m) ... (10) 

Then the values of X„ X'_j are obtained by equating to zero the coefficients of £ 2 ' +r in 

the equations 

» a * J -Z)- : J5 + .^ + Jli , -'-A = o (11) 

^F 6 + Z)-' 1 B+r a » 6 +J-F^A = o (12) 

It will then be found that the principal coefficients of \ h X'_ t derived from the left-hand 
member of (1 1) are respectively a, o, and from (12) o, a, while all the other unknowns 
have coefficients small compared with a. 

132. Method for Approximation. — Denote the difference between the iih and 
(■i — i)th approximation to any function Q by Q w . The first approximations to 
G-j H, A, B are obtained by neglecting all the unknowns in these four functions. For 
G-, II have m s as a factor, while u ;> . %u s occur elsewhere in A, 15 only when multiplied 
by 9--s , which also has m 2 as a factor. In (n), (12) all the unknowns except those 
we are considering are multiplied by terms which also contain the same factor. Hence 
G (1) , H (1) , A (1) , B (1) , \p\ X'_i (1) are obtained from the known terms in the equations 
of § 131. 

The second approximation is obtained by substituting u 6 a) for u b in the various 
formula?. Then « 6 (S) is determined from 



: « 6 '% — «5 a, «0> 



G (2) = M s ll) M + tt 5 ,1, tt 0> H« : 

•3w 5 a) . 5« -"^ d) 7Sr +-|m J G (a) + 3m 2 (i + 2 m)i)- 1 H®, 



'. 



B <2) = u 5 m Ss + u s ™$8 + 2(1+ m) AF- 1 A' 3) + 3ni 2 ( 1 + 2m)D- 1 H e >-f- 2 (i + 2m), 
« 5 «» . ar 1 + M5 a) (so-aC" 1 )-^" 1 B <2) +AI , - I A tS ' = o, 
lT^.&i- i +i^(s -a,i- 1 ) + D- l B' S) +l'F- l A^ = o. 

The further approximations proceed in a similar manner. 

Exceptions to this method will only occur when the divisors introduced by 
F" 1 , I)- 1 are such as to render it useless, or very tedious, owing to the number of 
approximations required. For such cases a special method is devised in § 138, below. 
133. Development ofV, A'. — These functions depend solely on the terms in co 2 , co h 
where (§8) 

o> 2 = |m 2 (« 2 a 2 +s 2 fl 2 )+im 2 Mi'6 2 — m 2 s 2 6 2 , 
a'w 3 = |m 2 (M 3 a 3 + s 3 a 3 ) + fm 2 (M 2 sc 3 + Ms a c 3 ) — sm 2 s a (MC 3 + sc 3 ). 

iHence 

1/ = w 2 a 2 + s 2 a 2 + I«*6 2 -|« 2 62 + ^" 1 {-^{uW& i +s?Da 2 )-%u / 8m i + i& i Db l ) 



+ *? {¥(w s a3 + s 3 a 3 ) + f(M 2 sc 3 +tts 2 c 3 )- 



■ ;:s 2 V«^ + «3j, 



-D^i-MuWuz+^Mz) 



+ 1 (ifoDc 3 + mWlj) - 1 s*(nDt 3 + sDl,)} 



A' = it?a 2 -s%+ -T {f(« 8 a3-s 3 a a )+;J(tt%C3-$ 2 MC 3 )-s 2 (w 3 -sc s ) 

(A 



'• v 



S^^w^^w^^I^^w^^^pwS^^^^^^^^^^^^^^^v™ 



5 8 Mr. Ernest W. I'.eowx. Theory of the 

The terms in these expressions are all known, for the German letters having 
suffix 2 contain ef as a factor, and those having the suffix 3 have a as a factor. These 
are expanded (§ 9) in powers of e', while M s =(« , ) 4 +(tt , ),+ (t« , ),+ (u%+ (m j ) , &c. 
The terms whose characteristics are of order 5 are chosen out to obtain I/, A'. 

1 "4. Development of Sf, 8(Z>f ). — The calculation of these functions is troublesome 
and a more detailed exposition is advisable. We have (§ 129) 



Sf = c„ 



oc dg oc 2 de 



cf 3 



where the partial derivatives refer only to the coefficients in f, and c , g are put for c, g. 
The terms containing c 2 g 2 , g/, g 4 are absent, since f t does not contain g. The new 
]>arts of c contained in c 4 are as yet unknown ; they will be determined in Section (ns. 
and meanwhile it may be noted that they are only present in the coefficients of £'' il - 
The expression for Sf arises (§ 129) from 

Sf = 8{D%us)—Bu . Ss-(i + 2m)(u$s-s$u) +Q} , 

wllGTG 

Q^B-^-iD^-B-^B'^ + D'o,,) (13) 

To find 3f 3 /3c put 



«„r l = 2/tj4", ih£ l = ^,J' h , 



where 



a = 3, 2 ; h = 3-0 ; q-q t = zi + r = p ; 

and where the summation signs refer to all the terms present. Then 

PO 
9f 3 = 2?'£ [{p i + qqi+(i + 2m)(q + q l )}i l ^ qi + {p , + 2(?,-(i + 2m)( 9 + g,)}^- 3 / J -J + -g^ 



3c 



dc 1 



■H-iH-J 



+ 



OC 



the summation sign referring to all possible values of p, q, q x . 
Since p = q-qi, the first factor may be written 

•* do 3c \3c e'e y 
Also 

S^^ + fi^,) = S(Si,M ±M6S.«)- 

Omitting the summation sign for the two values of a, as in § 131, and putting 



(0=3, 2; 
U = 3-a. 



E 
i:' 



= M^^ttA jy =S45« ±S^W, a 



the results can be symbolically expressed in the form 

in which the sign 5' denotes that in the formation of E the proper coefficient arising 
from d p /dc-dq°/dc is to be attached to the corresponding term in E. and similarly for 



■ 



mm. 



Motion of the Moon. 



59 



the other two terms. The above algebraical expression appears more complicated than 
it is in actual calculation, for the derivatives of p, q, q 1 are integers less than 4. 
By a quite similar investigation and with the same notation we obtain 



I- (Di z >) = - 22' f(i + m) f E + i?' DW + l? Pi . 

dc L dc dc 00 J 



The derivatives with respect to g are obtained in replacing d/'dc by d/cg in the 
above formula?. 

The same formula? serve for oi\/d c , o(D^)/dc, but they simplify. We have 
Qi = o. a=i, b = o, q =p — 2t±c, q 1 = o; the only terms present are those of 
characteristic e. 

Finally, in a similar manner, we obtain 



3 2 f, 



2(M S + S e U ), -W-^- = — 2(M e S -S e M ). 



d. 



135. Development of 3Q 3 /dc, aQ s /dg.— Differentiating (13) in the previous §, we 
obtain from the only possible combination of a, b, 

which, in connection with § 133, gives the required formula. 

For the derivative with respect to g of the first two terms of Q we have 

Now z only contains £ in the combination £ 2 <+'-_£-2<-t i Let j' denote the value of ^ 
when this expression is replaced by £" +T +£~ 2i ~ T . Then, since only the first multiple of 
g occurs in z 2 , z u 



d 



Hence 






c'g L eg 'J dg 



Section (ii). J"Ae New Parts of c. Terms -with Small Divisors. 

136. Determination of c^ — In those terms of the fifth order containing £ 2i±c a new 
part. c it of c arises for determination, and one of the unknown coefficients is indeter- 



i , 



ire 



minate ; the definition of the latter (§ 25) is such that the coefficients of £ ±c in « 6 £~ 
to be equal. One of the unknowns is thus replaced by c 4 , but the linear character of 
the equations is retained. The formulae for finding c 4 and the coefficient of £ c must, 
Royal Asteon. Soc, Vol. LVII. k 



60 Mr. Ernest W. Brown, Theory of the 

however, be quite differently formed from those for the other unknowns. The main 
reason for this is that in the case of £ ±0 we have 

since c is put for c after the operation F" 1 (§ 130) ; the divisor is therefore very small. 
If all powers of m had been included in F the divisor would have been zero and 
approximation impossible. Hence the formula; must be so arranged that F _1 is not 
present. 

Equations (10), (1 1), (12) of § 131 may be written 

B = B' + (i+m)F 1 A, B = iD2(M a s,-sX), A =-F(ms + « 2 ), 



giving 



A-— — F2)2(M a s 6 -* a w 1) ) + ^£_=o ... ... ... ... (14) 

2 + 2m i+m 

- us -z i +~ - — D2{u a s„-s a u b ) = o ... ... ... (15) 

i + m 2 + 2m 

equations which are free from the operator F _1 , since A, B' do not contain it. These 
two quantities contain c 4 . To isolate it put (see equations (9), (10)) 

A = A' + c 4 i^».+(r + 2m)2)->|-(2« 1 ')} 1 

[_ v C C ) 

Substituting in (14), (15) we obtain 

C4 r^k + I ! + 2 m + -?_ iD^hm,')] = -A' + .-l {^2(«A~ ««« t )-B"} ... (16) 
[_dc I 2 + 2m) cic J i+m 

o = -j«.-s 2 + -i-rB"-i2>2(M A -« a « 6 ) + ic 4 2>- l9 -(i« 1 ')] (17) 

i+m L cc -■ 

137. These are well adapted for solution by continued approximation. Equating 
the coefficients of £ c to zero in each of them, it is to be noticed that the unknowns in 
A', B", $(u a s b s a u b ) have the factor m 2 at least (§ 132). If such terms be neglected in 
the first approximation, (16) is a simple equation to find c 4 . When c 4 has been obtained 
Ao + Xo', =2X , is found from (17), since ~a(X + X / ) is the cofficient of £>, while the 
other unknowns have the factor nr. When c 4 , X have been obtained the remaining 
\„ X'_, are found by the ordinary method given in § 132. The first approximation 
completed, the resulting values are used for n 5 , s s to obtain a second approximation to 
c 4 from (16) and to X from (17), and then to the other unknowns from § 132, and 



so on. 



There is less disturbance of the computation sheets than would appear. The first 
approximation to A is found, exactly as in § 132, by omitting all unknowns. In that 



Motion of the .Moon, 



6 1 



to B wo omit all unknowns and the term 2(1 + m)-'-F- 1 A in tlio coefficient of £ c , and 
then all the quantities for (16), (17) are to hand. A difference occurs in the second 
approximation, due to the fact that 



■-S' " ( Z)f '')> 
3m 2 do 



as 9*i 



must be respectively included in H (2) , A (2) (§ 131) for the unknowns other than 

138. Small Divi-wrst. — The operators 7) -1 , F" 1 introduce small divisors in the 
cases of long-period and monthly terms, respectively ; each set of terms whose argu- 
ments differ by 21 involves one of the former or two of the latter. As the basis of 
the continued approximation was the ability to neglect terms having m- as a factor, 
approximation may become impossible, or is at least very slow, when the small divisor 
is of order m 2 . The method used for this case is the same as that of § 29. 

Taking the case of a long-period term, suppose that the corresponding coefficients 
arc X , V- The first approximation to the other A,., X'_ f is first obtained with X , X,/ 
considered as unknowns, so that they are expressed as linear functions of X , X ' and a 
known part. The first approximation to X , X ' is then obtained by using these values 
of X i; \'_ { in w s , a« 5 , &c, instead of neglecting them. The process thus leads to two 
new simultaneous linear equations for X , X '. When these are solved a substitution 
gives the first approximation to the other unknowns. In the application of this rule it 
was found sufficient to determine X ±1 , X' Tls in terms of X , X ', a second approximation 
giving the required accuracy, and the other coefficients being determined by the 
ordinary method. 

For the two monthly terms, corresponding to coefficients X () , XJ, \_ u X/, the same 
process was followed with respect to that pair of them, say X . X,/, which had a divisor 
containing m 2 as a factor : the divisor of the other pair has then only m as a factor. 



Section (iii.) The Equation for z. 

139. The non-homogeneous form was retained for s, since the greater part of the 
series-multiplications and many of the additions had already been obtained in com- 
puting the terms of lower orders. The method for z is, therefore, the same as in previous 
chapters. 

The known part of the expansion of Kzj-fi (equations (16), (18), § 20) is expressed 
in the form 

.^^"B.+^'JJ. + ^'B.+^'B^ with *," = -U-.S, 
r Po 

Then, using the notation, 

^(,7' 7> ~)=f( u : 8 > *)-r-( M o. »o> Po). 



6 2 Mr. Ernest W. Brown, Theory of the 

for brevity of expression, we have 

B 1 =[u l +s 1 ]-~(u , s ), 

JB i= [u 2 + S 2 — f(« 1 a +8 l 2 )-|« l S l +» 1 !! ]-T-(Mo,So ) Po)» 

B 3 =[U 3 +S 3 -|(M 1 M 2 + S 1 S 2 )-f(iM2 + M2Sl)+2«l% + M(V + S 1 3 )+¥(«^ 2 «l+% S l 2 )-| ;:; l ! ( M l+ S l)J 

-r( M 0! s o> Po)> 

^=[^ + 84 — f(2M 1 W 3 +M 2 2 +2SlS 3 +*3 2 ) — I( W i«3 + M 3*l+ M 2 S 2) + 2SiS 3 + »./ 

+ 3 /(Mi 2 M2+Sl J S 2 )+VX M l 2s 2+ M 2Sl 2 +2 M l S l M 2+2«hSlS2) — |»l 2 (M 2 + *2) — 5«l ;S 2(Ml+*i) 

Tlie terms duo to 12, are givc-n by u) 8) 

-A = m h {b 2 + :; * 7 («c s + «,)} , 

in which b 2 . c 3 , c, are expanded in powers of </ and respectively multiplied by those 
terms in 2, uz, sz which give characteristics of the fifth order. 

Finally, the part of -D 2 %z^ which depends on c-c , g-g is 

-2S'(c 2 ^+ g|?) Dz 3 - S fri - 2g,^l', 

with the notation for 5/ given in § 134, and for z 1 given in § 135. All the terms are 
known with the exception of g 4 ; this quantity is determined as in §§31, 73. 



■ 



m 

till 



Section (iv). Nature of the Computations. Tests for Accuracy. 

140. The Computations for u M s K . — The equations, prepared for computation, have 
been given in § 131. The principal part of the labour is the formation of the products 

u a u b , u a s„, 5u ti . Bs b , u a $s h , Dz a . Dz h , z a z b , (a = 4, 3 ; 6 = 5 — a) 

in which s b =u^ £,?,_,— — iW 

For calculating I/, A' (§ 133) we require the products 

"o M M U a %. «a »i» u a % ««) U<x u b «a 

where a + /^4 in the first three and a + b + c<4 in the last two. Most of these, 
however, are deducible by briefer multiplications from the results obtained in calcu- 
lating the fourth order terms for which the functions Xu a u b /u \ Zu a $ b /u v s 0l &c, were 
obtained ; the multiplications of these by u 2 , ti % &c., arc easy on account of the rapid 

convergence of all functions of it , s . 

For computing Sf, S(J>f) (§ 134) we require 

u a s b , s„ $u„ s 2 « u »i Dz 2 , w, s , s Bui ; (0=3, 2 ; 6=3— a) 

the first, third, and fifth of these had been previously obtained. The products in § 135 
which contain an accented z are deducible by mere changes of sign from those formed 
without an accent. 

Of the remaining calculations which do not consist of additions or subtractions 
there are multiplications of series by constant factors {e.g. G by -Jm 2 in A). The 
operation D consists in multiplying each coefficient by the corresponding index of £, 



is 



§ 



m 



m 



M, •/!'•/} of /A, .IA./.?7. 



63 



while the operations J9 : , J)~ 2 , F 1 consist of similar divisions. In many cases the 
factors are small and the divisors greater than unity, while the number of significant 
figures is small, so that these operations can frequently be done without the use of 
logarithms. It is, in fact, much easier to divide mentally by a number of. sav, three 
figures than to multiply by such a number. Crelle's or Tamborrel's tables might 
have been used for both operations. As a matter of fact these aids were not employed. 
I believe that Avhere the numbers to be multiplied consist of only three significant 
figures, and do not run at all consecutively, it takes less time to use a four-place table 
of logarithms (which can be mounted on a single card) than to be obliged to turn over 
the pages of a bulky volume of multiplication tables. 

141. The Computations for .:. — The processes were the same as those of previous 
chapters. Of the k"„ 7?* all but k" ,, B. had been previously found ; K". r is obtained 
from the brief series multiplications of z t by -^i</2p s , while all the products and most 
of the sums in ll t had been obtained in computing u,,. The only multiplications 
remaining were the k",72 5 _, (i=i, 2, 3, 4). and the uz in tS^:, most of which latter 
products were at hand. The long-period terms do not produce small divisions, and the 
initial number of places of decimals adopted was such that special methods for the 
monthly terms were not required. The approximations also were sufficiently rapid. 

142. Tests. — All multiplications and additions were tested by the addition test 
explained in Chap. I., Sect. viii. Practically every operation comes under one of 
these heads. The apparent exceptions are the operations I), D~ l , F -1 . For J) we have 



I'-r-Z 



(22t / i ( + r3,,. t ),'* f 



If we put £ = 1 we obtain a test easy to compute, since i is a small integer. The 
same process performed on the result tests for Z)~ ] and a slight extension enables us 
to test similarly for F -1 . 

This method works well for the details of the calculations. The calculations 
en manse can be verified by going back to the differential equations and using the above 
method for the operator D with £ = 1. This was not used very extensively on the 
fifth-order terms ; for reasons which will appear in the rest of this section. Such a test 
is very laborious and it seemed hardly necessary to make it. 

143. An indirect test in fact arises with every set of terms with a given 
characteristic. Each such set is u, s, and every other set in z has one (long-period) or 
two (monthly) terms whose coefficients possess small divisors, and the process of 
division by these is practically the last step in the determination of the coefficients. 
Xow, in the fifth-order terms the chief danger of error arises not in the detail* of the 
calculations, which can be tested by the method just described, but in the possible 
omission of a whole set of terms, or in the use of a wrong set, owing to the very large 
number of sets to be dealt with. It is just such errors that would be detected, for even 
though they might not be very large the numbers which go to make up the long- 
period or monthly terms are always formed of the differences of numbers large in 



6 4 



Mr. Ernest W. Brown, Theory of the 



comparison with their algebraical sum, so that an error, after the process of division by 
a small divisor, may cause a coefficient to appear several times its actual value ; a 
rouo-h comparison with Delaunay's or Hansen's results would reveal the error at once. 
It must be remembered that such errors would practically run through the whole series 
of calculations, but would be mainly revealed by a few terms. Thus one of the chief 
causes of the extent of the calculations furnishes the most valuable test of their 
accuracy. Even if Delaunay's results were not available the test would often still 
work, owing to the fact that negative powers of m cannot be present, and therefore that 
the maximum order of magnitude of any coefficient can be roughly stated in advance.* 

144. A Special Set of Tests. — In a paper f published in 1896 I gave some extensions 
of Adams's theorems which connected the mean motions of the perigee and node, not only 
with the constant term of the parallax but with one another. In fact, equations were 
obtained relating the parts of c, characteristics k 2 , e 2 k 2 , k 4 , e' 2 k 2 respectively, with those 
of £r. characteristics e 2 , e 4 , e 2 k 2 , e¥ 2 . The former set is determined in connection with 
u from the equations for u, s (§136), the latter set from the non-homogeneous 
equation for z. There were thus four separate tests from these relations. Further, the 
coefficients of certain characteristics in the constant term in the expression for the 
parallax are related by separate equations to parts of the motions of the perigee and 
node ; these characteristics in the parallax are e 2 , k 2 , e 4 , eV 2 , kV 2 , k 4 , e 2 k 2 , furnishing 
seven other tests which, however, with the exception of the first two, are not 
very searching, owing to the small number of places of decimals used in the results for 
the fourth-order characteristics of the parallax. 

The relations in question are included in the equations 

do, , a [E+mv> _ 



I 1 - 



2 de 2 \ 






(18) 



(i9) 



where ft, /3 3 , /3 3 (c 1; c 2 , c 3 in my previous papers) is the set of canonical constants 
complementary to the constants giving the position of the mean Moon, its perigee and 
its node at time t=o ; tt u B x are the mean motions of the perigee and node ; and (Q)° 
represents the mean value of a function Q. 

145. I have already developed $ a method for the calculation of &, /3 3 solely from 
the coefficients in u, s, 



ft-2 



The following are the results 

■1 1844 44e 2 — -02324e 4 — -26363e 2 k 2 — -oonoeV 2 , 



na 



'■\— —2-00205 9k 2 ~i-96376k 1 — -28546e 2 k 2 — -oo568e' 2 k 2 . 



* An error, the use of u^~ l instead of w 4 /« in B 4 , was actually first revealed in this way, although 
the difference between these two series is divisible by m 2 . 

f " On Certain Properties of the Mean Motions," <fec, Proc. Lond. Math. Soc. vol. 28. 

X " On the Formation of the Derivatives of the Lunar Coordinates with Eespect to the Elements," 
Trans. Amer, Math. Soc. vol. 4 (1903), pp. 234-248. 



Motion of the Moon. 



65 



Also 



\ 1 +111/' V i+m/ 



The values of c , g„ are given in Chapter III., those of c 3 , g 2 in Chapter V., those 
of c 4 , g 4 in the following Section. They are 

c= + 1-07158 32774 + -00268 57ie 2 — -03465 6oe' 2 + -o538s 595k 2 --o22i2 6a 2 

+ , ooo23e 1 + -oi8ic 2 e' s + , ooi45e 2 k 2 + -i77oe' , k ! + -07657k 4 , 
g= +ro85i7 i4266 + -oo3i8 6i83e 2 + 'oos64 6535e' 2 - -00806 6255k 2 + -omo 58a 8 

+ -ooo2 7e 4 + -o 1 04e V 2 + 008 7 5e s k 2 — -oo9oe' 2 k 2 — -00883k 4 . 

146. Differentiate (18) with respect to k 2 , and (19) with respect to e 2 ; the two 
left-hand members should then be equal. Calculate the terms with characteristics 1, 
e 8 , k 2 , c' 2 in these members ; the results, which should be the same, give the following 
differences : 

(/3j) e *e k a = — -oo6-?7 804} ■>.„, 

V-^/e K 4 ^'f I (Jiff. = -00000 002 J 

(As)k=ge= = —-00637 892) 

(Pi) "k»c.. + (/3 3 ) e = ka g e= + 2(/? 3 ) k =g 1! . = --00291 1 diff< _ , ooooi . 
2(/3 , 2) a <o ka + (/3 2 ) «c . ka = — •00290J 

(/3s)e»k»Ck» + (/3 3 )e=k»g k » + 2(/3 !! ) e ,C lt , = - "03252 | ^g. __ . QQQ0I , 
2 (A))k«ge' + (fia)vg*°V = ~ -03251 ) 

W^ + (C^«- = — °»8}diff. = . O002 • 

(PsWge'-KfrlVgeV = — -0226) 

here (/3 2 ) e . denotes the coeflicient of e 2 in fi-,/nd 2 , &c. 

These differences should properly be divided by 2, because (fi 3 ) k >, which is accu- 
rately determined, is very nearly equal to 2. 

The first result tests certain terms of the third order ; the other three results test 
terms of the fifth order. The latter involve a large proportion of the forms for 
computation of the fifth-order terms and of the results for the terms of the fourth and 
lower orders. 

147. Next, if we equate to zero the coefficients of 1, e 2 , k 2 , t' 2 in (18), (19), the 
coefficients of 1. e' 2 in the constant term of i/r should be zero, and, since E+ M = n 2 a z . 

i^K = ~(i +m)n« 3 Q° e . 1 '(&),, ,g k . = -(1 +m)wa 3 fy\, j 

From the results of the transformation to polars (Chapter IX. ) I find 

/ a \° 

I I = +"99999 97i + 'ooooo oe 2 + *ooi2 e' 2 + 'ooooo k 2 + -ooa 2 + oooi e 4 — -oooi eV 2 

+ •0040 e 2 k 2 + -ooo e' 2 k 2 — -0049 k 4 . 

It will be noticed that the differences from zero of the coefficients of e 2 . k 2 , eY 2 , t' 2 k 2 
do not exceed one unit in the last places calculated. A corresponding agreement will 
be found for the coefficients of e\ e 2 k 2 . k 4 . when calculated from (20). This decree of 
accuracy should be expected, since the number of places of decimals used in trans - 



(ao) 





^fBi 




66 Mr. Erxkst AY. Brow>~, Theory of the 


*'*°**llSl 



forming to parallax is smaller than that actually obtained in u, z. These tests apply 
to the results for the fourth and lower orders.* 



Section (v). Values of A, B, u£ l /a\ ; A, i~ A /a\. 

148. The following tables show the characteristics and arguments of the terms of 
the fifth order which have been calculated, together with the §§ in which the results 
are given. I have not set forth the types of coefficients, since these are sufficiently 
evident from the arguments and characteristics according to the scheme adopted in 
Section (iv), Chapter L, and illustrated in the later chapters. 



§ 


A 


Arguments. 


149 


e 5 


2t±5C, 2t±3C, 2i±c 


150 


eV 


2i±4C±mj 2-i±2c±m, 2i±m 


151 


eV* 


2i±30±2m, 2t±3C, 2i±c±2m, 2i±c 


152 


eV 3 


2*+ 20+301, 21+20+111, 2i±3m, 2i±m 


iS3 


e 3 k s 


2l±3C±2g, 2i±30, 2t±C±2g, 2*±C 


iS4 


eVk 3 


2i±2C±m±2g, 2i±2C±m, 2t±m±2g, 2i±m 


155 


ee'% 3 


2i±C±2Hl±2g, 2»+C+2m, 2l±C+2g, 21 + C 


156 


e' 3 k 2 


2i±3m±2g, 21+301, 2i+m+2g, 2i±m 


iS7 


ek* 


2i±c+:4g, 2i+e+2g, 2i+e 


158 


e'k> 


2t±m±4g, 2i±m±2g, 2i±m 


159 


e 4 <x 


2*^40, 2i!±2C, 2% x 


160 


e 3 e'a 


2"ii±3C±m, at^+c+m 


161 


e 2 k 2 a 


2t' l ±2C±2g, 2t 1 ±2C, 2li±2g, 21, 


162 


ee'k 2 a 


2i,±o±m±2g, 2t,±o±m 


163 


kV 


2-i]±4g> 2*l±2g» 2*1 I 


164 : 


k 5 


2i±Sg, 2i±3g, 2%±g J 


165 


k 3 e 2 


[ 

2 i±3g±20, 2i±3g, 2i±g±2C, 2*±g 1 


166 


k 3 ee' 


2t+3g+c+in, 2i+g+e+m 


167 


k 3 g /2 


2i±3g±2m, 2i±3g, 2i+g+2m, 2»+g 


168 


ke 4 


2t±g±4C, 2i±g±2C, 2t±g 


169 


keV 


2i±g±3C±m, 2i±g±c±m 


170 


keV 2 


2 *+g±2c+2m, 2»+g+2c, 2i+g±2rn, 2i+g 


171 


kee' 3 


2i±g±c±3m, 2i±g±c±m 


172 


k 3 ea 


2t,±3g±:c, 2i,±g±C 


173 


kVa 


2*i±3g ±m . 2i!±g±m 


174 


ke 3 a 


2t 1 ±g±3C, 2ii±g±C 


J 175 


keVa 


2i 1 ±g±20±m, 2i 1 ±g±m 



* On the subject of this section see two papers by the writer : " On the Degree of Accuracy of the 
New Lunar Theory," &c, Monthly Notices, April 1904 ; " On the Completion of the Solution of the Main 
Problem in the New Lunar Theory," ib., December 1904. 



Motion <>f tin .Vflon. 



The terms with the following arguments and characteristics required the method 
of § 138 to obtain the approximations with sufficient rapidity : — 



Arguments. 
;( 2 — c + 2m), 
:(c-2g), 

=(2 + C+2H1 — 2g), 

~(z — e + 2m — 2g), 

-1, 

:(i-c + m), 

:(i + 2C— 2g), 



Characteristics. 

eV 2 , ee' 2 k 2 

e 3 k 2 , ee' 2 k 2 , ek 4 

ee' ! k 2 

ee'% 2 

e ,4 «, e 2 k'-«, k 4 n 

eVa, ee'k 2 a 

e 2 k 2 a 

ee'k 2 a. 



±(i+c+m — 2g), 

For other terms where the approximations were slow it was found to be sufficient 
to calculate the third approximation, since a regular law of decrease then appeared 
which permitted the remainders to be written down from inspection. 

The results for A/a 2 \, B/a 2 \. u^jak are given in the following section under the 
columns headed A, B. u£~ l /ak. The last is a change of plan from previous chapters, 
intended to increase the clearness of the reading, since the results are always referred 
to their arguments and not to the special notations adopted for the coefficients. The 
choice of subsidiary results for publication at the present time was less easy with the 
homogeneous equations than with the ncn -homogeneous. It may be stated, in this 
connection, that long-period small divisors do not appear in A or B except in terms 
which have the explicit factor m 2 . while the monthly small divisors occur in B but not 
in A. Thus certain coefficients in the B-tablcs are not accurate to the last figure set 
down, but it is better to retain the same number of places for every coefficient under a 
given characteristic up to the final results. The work easily shows the extent to 
which the latter are correct, and in the tables for u^/ak none are given beyond this 
point. 

It is to be noted that A = A, B = --B, so that, given the coefficients in these 
functions for £ 2i+T , it is not necessary to write down those for £ Si ~ T . 

The coefficients of g in B are enclosed in square brackets, to signify that they are 
those parts of B denoted by B" in § 136. 

In Chap. VI. the zL-tables for the ^-equation were given in two parts (see § 98) ; 
here the parts are not separated, since the Z A arc generally quite small and easy to 
compute, and, further, they were not separated on the computation sheets. 

All the other arrangements of the tables are the same as those of Chap. VI. 



Royal Astron. Soc, Vol. LVIL 



68 



Mr. Eenest W. Brown, Theory of the 



149. Characteristic e 5 . Values of 



A 



B 



i. 


21 + 5c. 


2i+3C. 


2« + C. ' 


i. 


2S+5C. 


2J+3C. 


zi + c. 


4 






+ -00003 


4 








3 




+ •00007 


+ ■00094 


3 




— - OOOOI 


— '00014 




+ •00007 


+ •00246 


+ •01912 


2 


— 'OOOOI 


— •00035 


— '00486 


1 


+ •00256 


+ '05054 


— -OOII2 


1 


— -00036 


— '01208 


+ •00015 





+ •05446 


—•00888 


+ - OOOI9 





— •01258 


+ ■01396 


[+•01315] 


— 1 


— -02304 


— -00029 


+ '0002I 


— i 


+ •00824 


—•00093 


+ •00443 


— 2 


+ -00038 


— •00007 


+ -OOI24 


— 2 


— '00089 


+ •00051 


— '00222 





— -00003 


+ ■00047 


+ •00371 


-3 


+ ■00001 


— '00054 


— '00102 


— 4 


+ •00004 


+ •00039 


+ -OOOI9 


— 4 


— 'OOO04 


— -oooii 


— -00003 


Sum... 


+ •03444 


+ •04469 


+ ■02451 


Sum... 


—•00563 


+•00045 


+ "00946 



'-f-ae" 



21+ 5C. 



2J-SC. 



21+ 3C. 



4 ! 








+ •0001 














+ •0002 


+ •0001 


+ '0008 


2 






+ -0004 


— 'O002 


+ •0041 


— •0004 


1 


+ •0004 


— -0004 


+ •01 01 




—■0054 


— '0020 





+ •0103 


+ •0005 


— -0124 


+ •0010 


— '0042 


— •0042 


— 1 


— -0096 




+ •0030 


+ •0005 


+ •0036 


+ '0006 


— 2 


+ •0018 




+ '0001 




+ •0002 


+ ■0002 


— 3 










+ -ooot 




—4 














Sum... 


+ •0029 


+ •0001 


+ •0012 


+ •0016 


— •0015 


— •0050 



c . = +-00023. 



■ 



mi 

1 
m 



m 



m 

I 
IS 

1 

i 







































Motion of the Moon, 






69 








1 50. Characteristic eV. 


Yiiliu. 


s of 














A 














i. 


2i + 40 + m. 




21 + 40— m. 


21 + 2e + m. 






24 + 20 — m. 


2s + m. 






4 














+ ■00001 


— "00004 






3 






+ '00002 


— '00018 






+ ■00091 


— •OOI79 






2 


— *oooi8 




+ -00099 


— •00591 






+ •02674 


— •02614 






I 


-•00734 




+ •02727 


— •08748 






+ •23138 


— '03696 









— •10854 




+ •12325 


-•°7559 






+ •00218 


— '02908 






— I 


— •04499 




— -02582 


+ •00422 






— •02991 


+ •01844 






— 2 


— •00227 




+ 'oo 1 66 


+ •00869 






— -00282 


+ •10569 






~3 


+ •00130 




+ •00025 


+ ■01943 






— '00340 


+ •00957 






-4 


+ ■00134 




— •00016 


+ •00151 






—■00025 


+ •00029 




11 


Sum... 


-•16068 




+ •12746 


-•i353i 






+ ■22484 


+ •03998 














B 














«'. 


2s + 4c + m, 




21 + 40 — m. 


2% + 20 + m. 






2J + 2e-m. 


2i + m. 




.■: '■' 


4 






















3 








+ 'OOOOI 






— '00007 


+ "00027 






2 


+ '00003 




— •CCC.Gij 


+ •00091 






— •00433 


+ •00859 


I 




I 


+ '00112 




— '00420 


+ •02694 






-•08567 


+ -00043 


! 
1 




O 


+ •03275 




-•04343 


+•00945 






+ •02907 


+ •00054 


;;li;l;vll:lll 




— I 


+ ■01128 




+ - OI442 


-•00563 






+ •00055 


+ •02974 


1 




— 2 


— •00225 




— •OOOI5 


—■00336 






+ -00I2I 


— •04085 


j 


111 


— 3 


— '00119 




+ •01032 


— '00766 






+ -O0I30 


— ■00158 


1 


111* 


-4 


— •00050 




+ "00008 


— '00027 






+ 'OOO03 


— '00003 


! 


1111 


Sum... 


+ •04124 




-■033 5 


+ •02039 






-•0579I 


— •00289 


1 












M e v£ -1 -r a eV 














i. 


j 

I 2i + 4a + m. 

f 




2i— 4c— m. 




22 + 40-- a. 


2t — 4c + m. 






< 


t 






+ '0004 








— -oooi 






; 


* 1 






+ '6016 








+ -0003 




1111: 


- 








+ '0007 




+ 


0001 


+ - ooo6 






i 




— 


•0010 




— "0016 




+ '0032 


— 'ooog 






c 


) ■ 


— 


•0258 




— •0016 




+ •0322 


+ •0021 






— I 




— 


•0287 




— •0001 




— 


0216 


+ -0002 






~~ '. 




/: - : + 


•0103 








+ 


0013 








' 


: ' ; :> ; -* : //: 


+ 


•OQ03 








+ 


0001 








-4 


: ; ;::::::■ 






















Sum 


— 


•0449 




— - ooo6 




+ 


• OI 53 


+ •0022 





























mmMmmmlmSm^ 



7° 



Mr. Ernest ^Y. Brown, Theory of the 

..^-'-r-aeV (continued) 



i. 


2s + 2c + m. 


2»— 20-m. 


2*' + 20— m. 


2i — 2e + m. 


zi + m. 


2i— m. 


4 




+ •0002 




— •0001 




+ •0001 

i 


3 




+ •0060 


+ •0001 


— ■0010 


— '0003 


+ •0014 i 


2 


—•0009 


+ -0045 


+ -0036 


— •0014 


—•0072 


+ •0307 1 


1 


— ■0224 


—•0005 


+ •0626 


—•024s 


— •0207 


— •0062 1 





— •0472 


—•0051 


—0138 


+ - ooo8 


+ -0405 


— '0298 1 


— 1 


+ •0193 


—•0014 


+ •0294 


+ -0044 


+ •0084 


— '0060 1 


— 2 


+ •0024 




—•0005 


+ •0001 


+ •0022 


— "0005 ' 

! 


-3 
-4 


+ -0005 




— •0001 




+ •0001 




Sum... 


-•0483 


+ •0037 


+•0813 


—•0217 


+ •0230 


-•0103 1 



151. Characteristic e'V 2 . Values of 



i. 


21 + 30 + 2m. 


2S + 3c — 2m. 


2J+3C 


2t + c + 2m. 


2( + C-2m. 


2J + C. 


4 1 










+ •0001 




3 




+ •0001 






+•0055 


— •0019 


2 1 


+ •0002 


+ •0068 


— •0027 


—•0025 


+•1347 


-•0514 


1 


+ ■0030 


+•1315 


-■0631 


-•1787 


+•5806 


— •2882 


1 



— '0484 


+TS23 


-•1234 


-•0179 


—•0163 


+ •0170 


-1 


-•0537 


-•0035 


—•0079 


-•0638 


—•0038 


+ - 0235 




— •0271 


+ '0048 


— •0041 


+•2655 


-•0637 


-•0939 


-3 


+ '0345 


— •0063 


—•0078 


+ -0407 


—•0013 


-•0130 


-4 


+ '0038 


— "O00I 


— •0012 


+ •0014 




— -0004 


Sum... 


-•0877 


+•3156 


— '2I02 

B 


+•0447 


+•6358 


-•4083 












i *■ 


2i + 30 + 2m. 


24 + 30— 2m. 


2J + 30. 


2* + c + 2m. 


2i + o — 2m. 


2i + c. 


! 4 
1 3 


; :^0£§% 








— •0005 


+ -O002 


i 2 




— •0007 


+ '0002 


— •0016 


-•0181 


+ •0074 


1 1 


— '0013 


-•0205 


+ •0100 


-•0337 


— •0616 


+ •0859 





— •02IO 


— •0282 


+ •0540 


+ •0058 


-■0887 


[ + 'IS9S] 


— 1. 


-'003 1 


-•0235 


+ ■0361 


— •0189 


— •0424 


-'0473 


! — 2 


+ ■0120 


— •0039 


+ •0026 


-•0599 


— •0123 


+ •0308 


-3 


— 'OO9O 


— ■0014 


+ •0033 


-•0053 


— •0005 


+ •0017 


i -4 


— - 0004 


— •0001 




— -0002 






| Sum... 


— -0227 


-•0783 


+ •1062 


-•II38 


— '2241 


+ •2382 



Mutton of the Moon. 



7i 



•i_ .., : ^-aeV 





I. 
3 


24 + 3c + 2m. 


22 — 3c — 2m. 


24 + 30 -2m. 


21 — 30 + 2m. 


2i+3e. 


21-30. 






+ '0l6 




— •002 




— -004 




2 


1 


+ •033 


+ •001 


— •021 




+ •014 




i 




— ■013 


+ •016 


+ ■015 


— '007 


—•020 







— - 0I2 


— '002 • 


+ •075 


+ -008 


— -042 


?':\ :; ^*^S' :/•••. 




— 1 


-•113 




+ ■078 


+ "001 


— •046 






— j 


+ •029 




— "004 




— "010 






-3 


+ •002 














Sum... 


-•094 


+ •034 


+ ■166 


+ '001 


—•105 


-•013 




:::■■ t. 


2i+e + 2m. 


2J — c— 2m. 


28 + C— 2m. 


2*-0 + 2m. 


2» + e. 


2J-C. 




3 




+ •006 


+ ■001 






— •002 




2 


— ■001 


+•118 


+ •018 


— •020 


— •007 


-•O38 




1 


-■051 


+ •042 


+•235 


— '04 T 


— - I0I 


-•068 







— •063 


-•035 


+ '027 


+ •091 


— -xoo 


— 'I0O 




— 1 


+ •072 


— ■007 


-•094 


+ •028 


+ •080 


— •010 




— 2 


+ ■014 




-•003 


+ •001 


— "004 


— •001 




-3 
















Sum... 


— •029 


+ •124 


+■184 


+•059 


—■132 


— -219 










c e v = + 'oi8i. 









152. Characteristic eV 3 . Values of 



i. 


21 + 2C + 3m. 


22+2C — 3m. 


24 + 20 + m. 


2J+20— m. 


2«'+ 3m. 


2s + m. 


4 
3 




+ - ooo8 




—•0003 




+ •0001 


2 




+ •0289 


+ •0015 


— ■0148 


+ '0064 


-•0193 


1 


+ '0010 


+ •4412 


+•0039 


-•2390 


+ '0396 


+ •0623 





-•0588 


+•1698 


—•4221 


+ ■2731 


+ -0994 


+ - 399 2 


— 1 


+ ■6702 


-•0565 


-•1867 


+•4332 


+ 7555 


—•4246 


— 2 


+ •2282 


+ •0134 


—•0876 


-•0586 


+ '4i73 


-•1979 


-3 | 


+ •0813 


+ •0014 


-•0330 


-•0057 


+ '0192 


— •0088 


-4 j 


+ •0031 




— '0012 




+ "0005 


— 'O002 


Sum... 


+ ■9250 


+ •5990 


—•7252 


+ '3 8 79 


+ I- 3379 


— •1892 i 



72 



Mr. Ernest W. Brown, Theory of the 



B 



i. 


2 i + 2c + 3m. 


2* + 20 — 3m. 


24 + 2C + m. 


28 + 20 — m. 


28+ 3m. 


2t + m, 


4 
3 












— •0001 


2 




— •0017 


— -0004 


+ •0013 


+ "0004 


— •0030 


I 


— - ooo4 


+ •0242 


— •0069 


+ •0096 


+ "0046 


+ •0178 


O 


— "0291 


+ •0075 


-'"45 


+•1379 


+ •0278 


-•0457 


— I 


+ •043! 


+ -0086 


-•0156 


-■0348 


+•1365 


-•0423 


— 2 


+ ■1104 


+ •0038 


— •0287 


— '0223 


— -0023 


+ •0072 


-3 


— - OOII 


+ •0002 


+ •0010 


— •0010 


— - OOII 


+ •0006 


—4 


— •0002 




+ •0001 








Sum... 


+ •1227 


+ •0426 


-•1650 


+ •0907 


+•1659 


-■0655 



«w» 'C~ l — aeV 



i. 


28+20+ 3m. 


28 — 20 — 3m. 


24 + 20 — 3m. 


2J — 20 + 3m. 


2J + 2C + m. 


21 — 20 — m. 


3 




+ '012 








-•005 


2 




+ •260 


+ •003 


+ - 007 




— -091 


1 




+ •138 


+ •059 


— •OI7 




+ •064 





— "021 


— •005 


+ - I43 


+ ■045 


— •162 


-•°4S 


— 1 


-•537 




+ ■071 


+ - 004 


+ ■029 




— 2 
i 3 


+ •065 








-•030 




Sum... 


-"493 


+ '4°5 


+ •276 


+ 039 


-•163 


-•077 



i. 


2J+20 — m. 


24—20 + 111. 


2i + 3m. 


24— 3m. 


24 + m. 


2% — m. 


3 




— 'OOI 




+ •002 




— 'OOI 


2 


— 'OOI 


— •022 


+ •001 


+ ■058 


— - 002 


— -026 


1 


-•028 


+ •323 


+ - 02I 


+ •678 


+ '021 


— "269 





+•135 


+ •071 


— '224 


+ ■215 


-1-045 


+•782 


— 1 


— - 6io 


— 'OOI 


+ "220 


+ •003 


— '060 


+ •023 


— 2 


—•003 




+ -O03 




— 'OOI 




-3 














Sum... 


-•507 


+ ■370 


+ - 02I 


+ •956 


—1-087 


+ •509 



Motion of the. Moon. 
153. Characteristic : e 3 k 2 . Values of 



73, 



i. 


21 + 30 4- 2g. 


2J + 30-2g. 


21 -i- 3c. 


2;-.- r + 2-j;. 


2i'+C-2g. 


2« + C. 


' 4 










— "OOOOI 


— 'OOOOI 


3 




— 'OOOOI 


— "00002 


+ "00004 


— "00022 


— '00015 


2 


+ '0OOO2 


— "OOIIO 


— •OOOO7 


+ "00024 


-•OI654 


— -00488 


1 


+ '00099 


-•05196 


4- "oo 1 7 7 


-■01714 


+ '00407 


+•04333 





+ •00943 


+ •00377 


+ '42905 


-•839SI 


+ ■00270 


— '00006 


— 1 


-•09563 


— '00030 


— "00666 


+ •02984 


— •28820 


+ ■00854 


-> 


+ '00264 


— "02607 


+ •00274 


+ •00175 


— •OO762 


—02586 


-3 


— '00015 


— •00078 


— •00387 


— "00096 


— •0O0O2 


— '00207 


-4 


+ •00011 


+ "OOOOI 


— ■00021 


+ "00006 


+ '000OI 
-■3OS83 


— '00003 


Sum... 


1 —'08259 


— •07644 


+ •42273 


—•82598 


+ •01881 , 

1 


B 


!. i 2*'+3e + 2g. 


2J + 3C — 2g. 


28 + 3c. 


zi + e + 2g. 


28+C — 2g. 


2t + e. 


4 ' 








— 'OOOOI 




3 


— "OOOOI 




— '00002 


— '0004 1 


— "OOOIO 


2 | — -00005 


— "00086 


— •00014 


— •00185 


— •01289 


-■00579 ' 


1 —'00297 


— "02605 


— '00869 


— '07990 


+ '02099 


— -11843 


—'12680 

1 


— •01865 


— ■24098 


+•04525 


— '02231 


[ + •24621] 


— 1 ~-o<>,^9 


— '01191 


+ "04258 


+ ■01589 


+ •01792 


+ •02627 : 


— 2 ' —"00665 


+ "00234 


— •OO183 


— "00009 


— •01684 


— •02208 i 


-—3 — 'OOOII 


— 'OOI2I 


— •OOI76 


— "002 20 


— '00039 


— •OOI31 


— I — 'COO I J 


— "OOOOI 


— 'OOOII 


— "OOOo6 






Sum... — '07330 


—•05636 


-•21093 


— "02298 


-■01394 


+ •12477 j 






We*- t 1 ~ 


- ae 3 k 2 






i. 
4 • 


2i + 3e + 2g. 


2t-3C-2g. 


21 + 3C — 2g. 


28-30+2g. 


21 + 3C. 


2J-3C. 




+ •0001 








— "OOOI j 


3 i 




+ '0003 




— "OO02 




— '0020 


2 




+ "OOOI 


— *00O2 


— "0096 




— "0064 I 


1 




— •0212 


— "0164 


+ "0204 


+ "O00I 


+ "0003 





+ '0005 


+ '0414 


-■0332 


+ "0478 


+ "08l2 


+ ■0652 


— 1 


—•0292 


+ •0008 


+ •0034 


— "OOIO 


— •0035 


+ •0030 


— 2 


+ •0104 




+ - ooo6 




+ '00OI 




-3 
-4 


— '0003 




+ "0004 




+ "0001 


is;: ^o-;y;i- ;.|^ ;'.'•'• 1' 


Sum... 


-•0186 


+ •0215 


—•0454 


+ •0574 


+ •0780 


+ '0600 

1 



74 



Mr. Ernest W. Brown, Theory of the 



u eV £ _1 +ae 3 k 2 (continued) 





2J + C+2g. 


2i — C-2g, 


21 + C — 2g. 




2t — C + 2g. 


2J + C. 


zi — e. 


3 




— ■0003 


— •0001 








— •0005 


2 




— '0029 


— •OO57 




— •0017 


— •001 1 


— -0151 


I 


— '004I 


+ •0040 


— - oo6i 




-•0939 


— -onS 


-•0280 





-•2427 


— '0019 


+ •0612 




+ '2074 


— •0946 


— '0946 


— I 


+ -i3 r 9 


+ - 0253 


— •0025 




— •OO98 


+ •0570 


4-'0224 


— 2 


-•0053 


+ •0005 


+ ■0052 




-•O005 


+ •0019 


+ •0017 


-3 






+ •0001 






+ -0003 




Sum... 


— "1202 


+ •0247 


+ •0521 




+ •1015 


-•0483 


— '1141 








c oV = + 


00145, 









154. Characteristic : eVk 2 . Values of 
A 



f'. 




21 + 2C + m + 2g. 


2S + 2C + m — 2g. 


21 + 20 — m + 2g. 


24 + 2C — m — 2g. 


4 








+ •00001 : 


3 




— •00002 


+ '00001 


— '00005 • 


2 


i — '00003 


+ •00092 


+ '00015 


— •01053 


1 




— '00069 


+ •05303 


+ '00339 


— •16500 







+ ■01234 


+■38403 


+ •02137 


-■54035 


— 1 




— •22251 


-■25505 


— •00741 


+ ■01095 


— 2 




-•06377 


—•03567 


+ '00224 


+ •00588 


~3 




+ •00236 


— •00032 


— '00138 


+ •00006 


— 4 




+ •00055 


+ "00003 


— ■00016 




Sum ... 


-•2717s 


+ •14695 


+ •01821 


— •69903 


«. 


2t+ 2C + m. 


2J + 2C — m. 2J + ra + 2g. 


2!+m — 2g. 


28 + m. 


1 

! 4 






— '00003 


+ '00002 


1 3 


+ •00003 


— "OOOOI — 'OO0O2 


— •00056 


+ '00030 


2 


+ •00061 


— '00238 +'00019 


+ •00092 


+ '01034 


1 1 


+ •02468 


-■02687 +"03053 


+ •15664 


— '10067 


! 


+•42785 


— •47278 —-01030 


-•14557 


-•33o55 


1 - 1 


+ •10576 


—•44694 —'81041 


— -14192 


— -19920 


i — 2 


— -07460 


— •02770 —-03083 


— '00061 


— ■06463 


! -3 


-•01376 


+ ■00113 —'00003 


+ ■00007 


— •OO203 


; —4 


— '00034 


+ •00003 +'00007 




— •00007 


Sum ... 


+ •47023 


-•97552 --82080 


— '13106 


-73649 



■ 



HP 

mm 



mm 



Motion of the Mo»n. 



U 



75 



i. 


2i t- 2C + m + 2». 


2S + 2C + m — 2g. 


2S + 2C — m+2g. 


21 + 2C — m-2g. 




4 
3 




+ '00003 




— •00016 




! 2 


+ '00005 


+ •00234 


— •00037 


— -00707 




I 


+ -00285 


+ •05964 


— •01526 


— '11226 




!■■ o 


+ •06395 


+ "00685 


-•07843 


+•02514 




1 . — I' 


+ •04286 


+•05958 


— •02791 


+ '03346 




— 2 


—■00288 


-•03142 


+ ■00113 


+ ■00389 




-3 


-•00551 


— - ooio8 


+ ■00078 


+ '00015 




—4 


— ■00029 




+ '00004 






Sum 


+ ■10103 


+•09594 


— ■12002 


-•05685 





>*. 



4 

3 
2 
1 
o 

— 1 

— 2 

-3 

-4 



Sum 



2i + 2C + m. 



+ 'Q000I 
+ '00096 
+ ■05111 

+ •31953 

— •OI784 

-■ 6593 

-•OO757 

— '0OOI5 



+ •28012 



ji-jc-- in. 



— '0000 1 

— '00282 

— •12626 

-•31454 

+ ■01876 
— -00962 
+ •00186 

+ '00002 



-•4j26l 



2j + m + 2g. 



+ '00001 
+ '00132 
+ ■03039 

— ■ 10006 
+ •04470 

— ■06061 

— •00299 

— 'OOO06 



-■08730 



2i + m-~2g. 



+ •00078 
+ •01530 
+ '00096 

+ '161133 
■-■16277 

-•00835 

--'COOK) 



+ •01509 



21 + m, 



+ -00039 
+ ■01883 

— ■01694 
+ •02876 

— •20480 
-•05815 

— 'C0I2iS 



-•23319 



u eW . T'+acVk 2 



p" 


«. 


22 f 20 + m + 2g. 


2t — 2C-m — 2g. 


2i + 2c + m-2g. 2i- 


•ac— m + 2g. 


2l + 2C-m + 2g, 


28 — 2e + m — 2g, 


4 




+ '0001 










j|lli!lilfil?l:?; 


3 




+ •0035 




— •0001 




-•0015 


^^^S9mfi 


2 




+ "0043 


+ '0001 


-•0055 




— 'OOOI 


\ 


1 




— '0840 


+ '0258 


-•1856 


+ '0002 


— •0077 


Wgggg:; 





+ '0001 


— '0489 


+ ■1474 


— ■2308 


+ '0011 


+ •0676 


^S0llMfxfMx>ii. 


— 1 


—•0816 


— ■0013 


— •0327 


— ■0014 


— '0008 


+ ■0057 


mjm^M^M 


— 2 


+ '0500 




+ •0229 


— '0003 


— ■0049 


+ '0001 


pill 


-3 


+ '0007 




+ •0005 




— '0003 




-4 


+ '0001 












Sum ... 


—•0307 


— •1263 


+ ■1640 


-•4237 


— '0047 


+ '0641 


Hoy At AsTitox. 


Soc, Vol. 


LV1I. 




M 











r 



7 6 



Mr. Ernest W. Brown, 27teo^ 0/ tf<e 

w aVkJ ^-f-aeVk' (continued) 



i. 


2J + 2C — m — 2g. 


28 — 2C + m+2g. 


28 + 2C + H1. 


2i— 20— m. 


24 + 20 - m. 


2i — 2e + m. ■ 


4 
3 








— •0027 




.:• .' : : •: . ' : ;i 

4- '0002 ! 


2 


-■0015 


4- -0007 


+ •0001 


-■0851 


-—•0004 


— - OI26 i 


1 


— •1024 


+ '0154 


+ "0060 


— •0627 


— - oii4 


-•3298 1 





— -1113 


+ •2490 


4- -2517 


-•045S 


-■3S04 


4- -0044 1 


— 1 


+ ■0015 


— •0190 


— ■0154 


■—•0102 


+ •6405 


4--o2i6 j 


— 2 


— ■0027 


+ -0005 


4--oo68 


— '0002 


—•0052 


4- -0005 1 


-3 


— ■0001 




4-"0022 




—•0005 




-4 


:■' : ■-"■■■ ■' y y: : :S- : :i "■ ' y - 




: 4* 'OOOI 








Sum ... 


— ■2165 


+ •2466 


+•2515 


— •2064 


+ •2726 


-•3157 


»'. 


2i-j-m + 2g. 


2j— m — 2g. 


2i + m— 2g. 


2i— m + 2g. 


21 + m. 


2i — m. 


4 
3 






— •0001 




4- - oooi 


— '0003 i 


2 


+ - OOOI 


— -0240 


— ■0018 


— 'OO03 


4- -0019 


— •0127 


1. 


+ ■0032 


-•0536 


— -0170 


-•0174 


— •0014 


-■2568 





+ •0312 


— -0490 


-•0137 


-■1218 


4-7122 


-•5019 


— 1 


+ ■4615 


— ■0236 


4--H96 


— -0690 


4-'xo27 


-•OSSO 


—2 


-■0093 


— - ooo6 


+ ■0034 


4- -0008 


4- "0142 


— -0044 1 


-3 

—4 


+ •0003 






— '0001 


. 4- '0003. 


— -oooi 1 

1 


Sum ... 


+ ■4870 


-■1508 


4- -0904 


— •2078 


4- "8300 


-■8312 



155. Characteristic ee' 2 k 2 . Values of 
A 



»'. 


2i + c + 2m + 2g. 


2i + c + 2m— 2g. 


2i + c-2m + 2g. 


2i + c - 2m - 2g. 


3 






4- "oooi 


4- "0005 


2 


— •0001 


— -0008 


4- -0005 


-■0179 


1 


— ■O0O2 


+ ■0033 


4- -0105 


4--I473 





4- -0390 


— •0123 


+ •0582 


-•1332 


— 1 


— •3226 


— -2290 


4- 'ono 


— •oooi 


: ^- :-^~ 21- 


-•0546 


— •0024 


4- -0030 


— -0009 


-3 


4- - oi44 


4- "0003 


— •oooi 




Sum 


-•3241 


— •2409 


4- -0832 


—■0043 



■ 











Motion of the Moon. 






77 



o 

— I 

— 2 

-3 

Sum , 



31 - (" -r 2IM. 



— "0007 

— '0202 
+ •0625 
+ ■2677 

— ■2448 

— ■O078 

+ •0567 



21 + c — 2m. 



— "0003 

— -0181 
-■3772 
+ 7/>=6 

— -0326 
+ "coSo 

— ■0002 

-■0598 



21 ■ ■'. - 22. 



— - 0004 
+ "O003 
+ -OI37 
+ '0364 
+ •0312 

— - oo8o 
+ •0732 



21 ■•■ c - 2C- 
— ■OCC.} 

T- '0046 

-'°35 2 
+ ■0671 
+ •0867 
+ ■0018 



+ -IIS4 



2t +C. 

+ '0002 
+ ■0078 
+ •2580 

+ I464 
+ •1292 

+ - OSOS 
+ 'CC2I 

+ "5942 



2f + + 2m + 22. 



— "0004 

— ■0165 

-■3784 

—•0444 
-•0052 



■■4449 



B 

22 + C + 2m — 2g. 

— '0009 

—•0817 

-•0493 
+ •0744 

— "0089 

— '0001 
-•0665 



2t + c — 2m + 2g. 


2t + C — 2m — 2g. 




— -0007 


— 'OOIO 


— '0204 


— "0272 


-■6451 


+ '1033 


— •2671 


+ 1-3216 


— •0042 


+ "0107 


— '0002 


— '0003 




+ 1-4071 


-"9377 


' 


— , _J 



:/ : : :i,:: 


2t + C+2m. 


28 + — 2m. 


2! + e + 2g. 


2» + C — 2g. 


2t + C. 


3 








+ "0002 




2 


— '0002 


- # oi37 




+ -OI2S 


+ •0077 


1 


+ -OI72 


- -4509 


+ •0081 


+ -3700 


+ •2878 





+ •2883 


— 1-2136 


— -0228 


+ I -4800 


[-•070S] 


— 1 


-'9542 


— -8189 


-•0058 


— 'O284 


— '4794 


— 2 


-•l8 5 


+ 'o°53 


+ -0070 


+ "0020 


+ •0924 


-3 


-•OO39 


— 'OOOl 


+ •0029 




+ -00I9 


Sum ... 


-•8378 


—2-4919 


— -0106 


+ 1-8363 


— -1601 



7 8 



Mr. Ernest W. Brown, Theory of the 



Mo.-'=k=r' I -r-aee' 2 k 2 



i. 


2t + C+2IU+2g. 


2! — c — 2m — 2g 


2! + C + 2m- 


-2g- 


21--C — 2m + 


2g- 


2J + 0-2m + 2g. 


21 


c + 2m — 2g. 




3 




+ '002 


















2 




+ ■059 






— - 00I 








+ ■014 




I 




— - 242 


-•057 




—•030 




+ 'OOI 




-•32 




O 




•-•037 


-•031 




+ '036 




+ •004 




+•087 




— I 


— '240 




+ '333 




+ '038 




+ •27 




+ '022 




— 2 


+ •157 




+ •010 








— •090 








-3 


+ '002 




















Sum ... 


— •081 


— •2l8 


+ •255 




+ •043 




+•185 




-•197 





2J + — m 2g. 2i-c + m + 2g. 



21 + + 2m. 



2i — c — 2m. 



2* - c + 2m. 



3 








-> 


— '002 






1 


—•361 


— '001 


— 'OOI 





+ •070 


— '°49 


+ • 1 6 1 


— 1 


—•098 


+ i'z49 


+ 1 "006 


— 2 


— •001 


+ '007 


+ '072 


— 3 






+ "ooi 


Sum ... 


-•392 


+ I - 206 


+ 1*239 



•043 
•776 

•5J5 
•010 



-1 '345 



•002 

•073 
■898 
•892 
•008 



-i'S73 



+ "003 
+ "34i 
+ i'433 
+ -n6 
+ -003 



+ 1-896 



Sum 



21 + C+ 2$ 



21 — C— 22. 



21 + C. 



I 









— I 


+ •086 


— 2 


— •038 


-3 


— 'OOI 



+ •047 



+ •034 



-i'i76 



•401 



— - 002 










— 'o6i 


— 'OOI 




+ "OOI 


+ •008 


+ •098 


+ -167 


4- '009 


+ "038 


— ■164 


+ ■006 


-1-280 


+ •421 


+ '199 


+ •199 


—-'007 


— -060 


— •827 


+ ro 3 8 


— •071 




~ -002 


•—•004 


- -030 


— -ooi 








— "OOI 





+ 1-245 



— -029 



<W,= + -i77o. 



k 



Motion of the Moon. 



79 



156. Characteristic e /3 k 2 . Values of 

A ■ . 



siSiM 


2i + 3m + 2g. 


2t + 3m-2g. 


2i + 3m. 


2i + m + 2g. 


21 J- m — 2g. 


2t + m. 


3 










+ '0004 


— 'OOOI 


2 




+ •0018 


+ '0002 


+ "0002 


+ •0025 


— •0027 


1 


+ ■0004 


+ '0349 


— - oo8o 


+ •0017 


— •O436 


— '0190 





+ •1256 


+ ■1251 


— '2031 


+ •0174 


+ •0292 


+•1769 


— 1 


+ •3169 


+ •0186 


- -9697 


— -1714 


•0000 


+ •6411 


— 2 


+ •0920 


+ •0011 


- '0504 


— •0850 


— "0004 


+•0253 


-3 


+ •0047 


+ •0001 


— '00 1 1 


— '0029 




+ -0005 


urn... 


+ -S396 


+ ■1816 


— I'232I 


— •2400 


— '0119 


+ •8220 



B 



1. 


zi + 3m + 2g. 


2J + 3m-ag. 


Zi + 3m. 


2S + m+2g. 


2i + m— 2g, 


zi + m. 


3 

2 


: ; : : / :V-::: ; : : \;:: ■ 


— ■0021 


— - 000I 


+ •0001 


— ■0037 


— •0013 


1 


— "0002 


— •0187 


— '0184 


— '0003 


+ '0201 


— "0116 i 





— •0698 


+ •1567 


+ -1648 


-•1380 


+ ■1626 


—1038 


j — 1 


-•2423 


+ •0103 


-i - 5943 


+ •1286 


-•OO57 


+ ■8841 


; —2 


— - 02l6 


—•0006 


— -0320 


+ •0319 




+ ■0210 ! 


: -3 


-—■OOO8 




— "0004 


+ - ooo8 




+ "0002 


Sum... 


-'3347 


+ ■1456 


— 1-4804 


+ ■0231 


+ T733 


+ 7886 | 









v vv i~ 1 ^ 


■ae 


3 k 3 








I. 


21 + 3m + 2g. 


2s-3m-2g. 


2t + 3m-2g. 




2i-3m + 2g. 


24+ 3m. 


2J-3m. 


3 




+ •001 














2 




+ ■021 












— "006 1 


1 




+ •258 


+ •036 






+ '001 




— •203 





+ '002 


+ •02 7 


— MI 7 






+ •009 


+ '204 


—388 1 


— 1 


-•637 




+ •054 






— 'OIQ 


+ 1-289 


+ •003 


~*2 

3 


+ - 034 




+ •001 






+ '002 


+ - 0I0 




Sum... 


— - 60I 


+ •307 


— -026 






— •007 


+ i'5°3 


-•594 1 
1 



Mr. Ernest W. Brown, Theory of the 



n 



w^sj-^'-j-ae^k 2 (continued) 



i. 


2i + m + 2g. 


2i— m — 2g. 


2t + m — 2g. 


2i— m + 2g. 


2i + m. 


2t — m. 


3 

2 




— 'Ol6 








+ •003 ] j 


I 




-■380 


— •016 




— ■002 


+ •097 ' | 


O 




+ •110 


-■150 


— -001 


+ •091 


+ •014 j ! 


— I 

— 2 


+ •586 
-•031 




— •018 


+ ■027 
+ ■003 


— •688 

— ■006 


+ •003 | 


-3 














Sum... 


+ •555 


-•286 


-•184 


+ •029 


— '605 


+ ""7 5 



Sum.. 



+ •00405 



157. Characteristic ek 4 . Values of 

A 



i. 


2t + C + 4g. 


2i + c-4g. 


2i + C + 2g. 


21 + C— 2g. 


2« + e. 1 


3 




— - oooo8 


+ - OOOOI 


— '00003 


— •00002 


2 




+ •00093 


+ -00005 


+ -OOI3I 


+ •00007 


1 


+ ■00002 


+ ■02617 


+ •00443 


— •02682 


— -01225 





— '00024 


- -04835 


+ - 03463 


— •26771 


+ •05994 


— 1 


+ •00379 


— -00499 


-•OOO95 


+ '0I4l6 


+ •02155 


— 2 


+ •00061 


— '00007 


+ - 003I3 


— •OOO33 


+ •00218 


; -3 


— ■00013 




— •00075 


— •00002 


+ •00013 



— '02639 



+ •04055 



—•27944 



+ ■07160 



B 



i. 


2% + c + 4S. 


2i + c-4g. 


2t + + 2g. 


21 + C — 2g. 


21 + C. j 


3 




— '00010 






— 'OOOOI 


2 




+ •00177 


— 'OOOOI 


— -oooi I 


— -00058 


1 


— '00004 


+ '09268 


— -00427 


+ -18160 


- -04036 





— '00213 


+ •00898 


— •80476 


-3 -I 5 6 94 


[-1-04073] 


-1 


+ ■04491 


— •00200 


+ ■36829 


— -16987 


+ '29027 


— 2 


— ■00518 


— 'O00O2 


+ "00192 


— '00136 


— '00736 


-3 


— •00017 




— "00036 


— 'OOOOI 


— '00008 


Sum... 


+ '°3739 


+ -I0I3I 


-•43919 


—3-14669 


- 79885 



Motion of the Moon. 

tW^-T-aek* 



Si 





i. 


21 + C + 4g; 


2i-c-4g. 


24 + C — 4g. 


2J — C + 4g. 




3 




+ '0002 








2 




4- "0009 


— ■0028 






I 




— -0604 


+ '0207 


— 'OOOI 




O 




+ •0014 


-'4579 


— •0027 




— I 






— '0024 


+•0959 




— 2 


+ - 0082 






— •0077 




i -3 


— '0002 








^^Pil|; : 


Sum 


+ "0080 


-■OS79 


-•4424 


+ -0854 




i. 


ai+e + 2g 


2t~C — 2g. 


2t + C — 2g. 


2%— e + 2g. 2j + e. 


2% — e. 




3 


• 


• !'— 'OQo'i. •'■ 








^^Blii% 


' 2 


—•0074 


— "00 1 1 




+ '0004 


llilS^"lll|K$$ftK "$;■;'' 


I [ 


-•0034 


— '2171 


— '0074 +'0004 


-•0805 




O 


+ •0013 


+■48x3 


+ 4-S387 


— 1-9841 H--3342 


+ ■3342 




— I 


+ '0032 


+ '0002 


— -0484 


+ -7232 + 'ioo8 


— -0109 




— 2 

-3 


—•0089 
+ •0001 

-■0043 




— '0008 


— -0017 — -0008 


— 'OOOI 


||p||l|fi!!tl>§x'l 


Sum... 


+•4705 


+ 4-2713 


— 1-2700 +-4346 


+ ■2431 










c u . = +'07657 


.: : ■ < :i v : :' i:: . . : : ' : ; ; : : ' ; ■■>-[ 





158. Characteristic e'k 4 . Values of 

A 



i. 
3 


2i + m+4g. 


2t + m— 4g. 


2s + m + 2g. 


2t + m — 2g. 


28 + m. | 




— •00015 


— -OOOOI 


+ -00006 


— •00002 


2 




+ •00730 


+ -00006 


+ •00526 


— •00076 


■ ■ ■ 1. 


+ •00002 


— "0I2I2 


+ •00133 


—•04200 


— -00520 





— '00025 


+ -00I55 


+ •03301 


+ •00136 


— •05614 


— 1 


+ ■04376 


+ "00005 


+ "02084 


— •00414 


- -05545 


—2 


— ■0081 1 




— -01413 


— -00015 


+ •00265 


-3 


+ "00034 




— ■00039 




+ '00003 


Sum... 


+ •03576 


— •OO337 


+ •04071 


— ■03961 


-•11489 



82 



Mr. Ernest W. Brown, Theory of the 
B 



»'. 


2s + m + 4g. 


2j' + m-4g. 


24 + m + 2g. 


2t + m — 2g. 


2i + m. 


3 




+ ■00019 




+ •00002 


— -ooooi 


2 




— -00464 


+ '00008 


+ •00316 


—•00139 


I 




+ •05167 


+ •00825 


+ •00155 


-•09713 


O 


— •OOI57 


+ •00266 


+ "45! 95 


-•50959 


— •03209 


— I 


— -09909 


— -00005 


+ •08329 


-•03351 


+ •22527 


— 2 


+ •00821 




— '01241 


— ■00034 


+ •00230 


-3 


— •00042 




— •00013 


— -ooooi 


+ •00001 


Sum... 


— •09287 


+ •04983 


+ •53103 


-•53872 


+ •09696 







w A .f" 1 -T-ae'k* 






«'. 


2t + m + 4g. 


2i — m — 4g. 


2j + m-4g. 


2i-m + 4g. 


3 




+ •0002 


— •0001 




2 




+ "0030 


+ •0013 




I 




— -1301 


+ •0771 









+ •0020 


+ •0026 




— I 


+ "0007 




+ •0001 


— - ooo6 


— 2 


+ •0198 






—•0154 


-3 


+ -0003 






—•0002 


Sum 


+ •0208 


— •1249 


+ •0810 


— -0162 



t. 


24 + m+2g. 


2t— m-2g. 


2! + m~2g. 


2t— m + 2g. 


21' + m. 


2i— m. 


3 




— -0001 










2 




— -0040 


+ •0016 




— '0001 


+•0003 


1 




— •1026 


+ ■1115 


— •0002 


— ■0044 


+ •0036 





+ •0011 


+•1848 


-•1833 


— •0024 


—•1037 


+•0735 


— 1 


+ •0472 


+ •0022 


-•0068 


-•0650 


+•1338 


-•0387 


— 2 


+•0133 




— •0001 


— -0064 


+ '0002 




-3 


+ •0001 












Sum... 


+ •0617 


+ "0803 


-•0771 


-•0740 


+ •0258 


+•0387 



Motion of the Moon. 



159, Characteristic e*a. Values of 



83 



21. 


21 + 4c. 


21 + 20. 


zi. 1 


1 7 




+ '0001 


+ 'OOOI ! 


5 


+ "OOOI 


+ •0013 


-•0314 


3 


+ '0022 


- '°794 


-•1150 


i 1 


— •0670 


-•0931 


+ •0027 


— r 


+ •'415 


+ •0106 




-3 


+ - 004I 


+ -0098 




1 — s 


+ •0025 


— •0320 




! -7 


— 'OO29 


-•0054 


1 


Sum | 


+ "0805 


-•1881 


—■1436 



J 21. 

7 


2% + 4c. 


2J + 20. 


2i. 






+ •0001 


1 5 




+ 'OOOI 


+ '°°45 


3 


+ "O002 


+ •0138 


— 'OOOI 


1 


+ -OIS2 


— 'OIIO 


+ •0238 \ 


— 1 


— •O277 


— - oi7i 




3 


+ "OO08 


+ '0002 




— 5 


— - OOI2 


+ '0004 




7 


+ '0001 


+ '0004 




Sum 


— •0126 


-■013? 


+ •0283 



'V.f''+ae 4 a 



/ 

5 
3 

1 

-1 

6 

"S 

-7 



Sum 



24 + 40. 



— ■012 
+ •051 
+ •008 
-•003 



+ '044 



28 — 4c, 

— '001 

— '0Q4 
+ '002 
+ -003 



'000 



21 + 20. 



— •OI3 

— ■O23 
+ •087 

— ■008 

— '002 



+ "04I 



28 — 20, 


2t. ••. 




— 'OO I 




■ : i 


— •OI3 


— '005 


> 


— •OI3 


— "041 




+ ■036 


+ '020 


''.;| 


-•005 


-■095 




— "001 


— •006 




+ •003 


-■127 





1 60. Characteristic eh' a. Values of 

'A 



■ ■ zi. ■'■■■■.■"■■■ 


! 28 + 30 + m. 


21 + 30— m. 


2i + c+.m. 


2t + c — m. 


•7 








.+ '0009 


.5 


1 — 'ooor 


+ '0015 


+ "OIIO 


+ '0170 


' '3 '; 


| ' +'oi6i 


+ -0275 


+ 'SSoz 


— "4322 


I 


+ 7904 


- -1771 


+ 2-5540 


- '1684 


— I 


1 --0381 


-i"3735 


+ 557° 


+ i'2i47 


_ 3 


I -'2193 


— '5492 


— -2981; 


+ '3211 


-s 


— ■0708 


— -0196 


- -1847 


+ -1151 


— 7 


— ■0196 


+ '0083 


+ -0034 


+ '0029 


Sum 


+ •4586 


— 2'082I 


+ 3"i936 


. +I'07M 



Royal Astkox. Soc, Vol. LVII. 



84 



Mr. Ernest W. Brown, Theory of the 



Sum 



2!. 


2i + 3c + m. 


1 




' 5 




3 


-'0059 


i 


— •2621 


— i 


+ •1203 


_ 3 


-•0134 


1 — 5 


— •0220 


— 7 


+ •0013 



-•1818 



2i + 3e — m. 



+ '0034 
+ ■0198 

-'3479 
+ -0400 

— ■0180 

— '0022 



-•3049 



— ■0031 
-•I368 

+ '8794 

— •0729 

— ■O23O 
+ ■0124 
+ '0006 



+ •6566 



+ ■0029 

+ '034I 

— •O464 

— '1214 

+ •1002 
-■0288 

— •0007 



— '060I 



« eV „£ -1 -*- ae3e ' a 



2i. 


2t + 3c + m. 


2t — 3c — m. 


2S + 3C-M. 


28 — 3c + m. 


2s + c + m. 


21 — e-m. 


28 + c — m. 


21 — c + m. 


7 




— •004 




+ •002 




— 'OOI 




+ "OOI 


s 




-■065 




— •on 


+ -003 


— •032 


— '002 


+ -027 


3 


+ •005 


-■056 


— •001 


— -116 


+ -130 


-•238 


- - o73 


+ -179 


1 


+ •183 


+ •003 


-■039 


— •190 


+ 1-064 


— '106 


- -171 


+ i'95 


— 1 


-•055 


+ '0I3 


— •668 


— •004 


- -286 


+ •374 


— 2-41 


- -030 


— 3 


+ •150 




+ '457 


— •001 


— "072 


+ •009 


+ -033 


— -004 


-5 


— •014 




— '004 




— '002 




+ '002 


— 'OOI 


-7 


— •001 
















Sum 


+ •268 


— •109 


-•255 


-•320 


+ -837 


+ '006 


— 2'62I 


+ 2T22 



161. Characteristic eVa. Values of 

A 



21. ; 


2t+2C + 2g. 


2t + 20-2g. 


21 + 2C. 


2 i + 2g. 


28. 




7 

5 










+ •0001 






— •0043 




— '0007 


+ •0026 




3 


— •OOOI 


— •02 72 


+ '0271 


-•0243 


+•5215 




1 


+ •0304 


+ -0033 


+ 1-0340 


+ •4413 


-■1873 




— 1 


+ '3747 


4- -0179 


- -1647 


+•0615 






-3 


+ -0S40 


+ •1558 


- '1047 


-•0183 






-5 


+ •0111 


-•0058 


+ -08l3 


— •0200 






-7 


— '0040 


— '0002 


— -0006 


— - ooii: 






Sum 


+ •4661 


+ 'I395 


+ -8724 


+ '4384 


+ •3369 





Mutivn of flu- Mam. 



S.s 



I 21. j 


21 + 20 + 2g. 


24 + 20 — 2g. 


2 i + 2C. 


2» + 2g. 


21. 


7 1 










+ •0001 1 


1 S ! 




4- 'O002 




+ '0003 


+ •0067 i 


l 3 


+ '0013 


+ '0212 


fOI38 


+ •0435 


+ •2328 1 


I 


+ ■0956 


+ •0894 


+ -40IO 


-^SSo 


+ •;-'>$ 


— I i 


+ '04IO 


-•07IS 


— •5640 


+ ■2047 




I ~3 


+ ■0415 


— •O32O 


+ ■1863 


+ •1826 




1 ~5 ! 


+ "004c; 


+ -0O49 


+ ■0339 


+ -0079 




7 . 


+ "0006 


— 'OOOI 


+ '0005 






Sura... i 


+ •1849 


+ '0121 


+ •0715 


+ '3°40 


-f-*96Sy 













-aeWa 










21. 


2i'+20 + 2g. 


21 — 20 - 2g. 


2% + 20 — 2g. 


21 — 20 + 2g. 


2i + 2e. 


2» - 2C. 


2l'+2g. 


28 — 2g. 


2%. j 


■ 7 




— "002 
















5 




— •014 








+ ■027 




— •on 


+ -003 


i 3 




+ •027 


— •021 


+ '°33 


+ -007 


+ •269 


+ '002 


+ ■066 


+ -iSl) ' 


J 1 


+ •002 


+ •062 


— •069 


-'°S3 


+ '335 


+ •501 


+ ■151 


-•166 


+ -948 


1 -I 


+ -III 


— •030 


+ •162 


-•150 


-1-251 


— •021 


+ •010 


— -061 


-1-693 


-3 


+ ■050 




-•048 


— •005 


— '054 


— •OO3 


—•362 


— •016 


— *026 


1 ~s 


— •024 




— •002 




— "007 




—•003 




— "001 


1 ~~ 7 
Sum... 


— •001 


















+•138 


+ - 043 


+ '022 


-•175 


- "97o 


+ 773 


— -202 


-•18S 


- -580 



7 
5 
3 

j ' 
— 1 

• -s 

7 



162, Characteristic efk* a. Values of 

A. 



2t+c + m + 2g. 2t' + c+m— 2g. 2i+c—m+2g. 2» + e— m— ag. 



+ •0002 
+ •0065 
+ •0766 

— ■0163 
-■3091 

— -1008 

— "0028 



+ '0070 

+ -0457 
—2-3608 

— '2389 

— -0729 

— - 0020 



+ *000I 

+•0055 

+•1939 
—7781 

+ •0376 

+'0530 

+ "0009 



— -0003 

~ '°333 

— -0382 

+ -1489 
-1-7625 
+ •0011 
+ "0014 
+ -oooi 



2» + c + m. 



— '0010 

— -137-1 

■-5-9398 

— "2096 
+ 1-2745 

— -0282 

— -0009 



— -oooi 

— '0019 
+ -0678 
+ 1-3612 

—4-0717 

— 1-4012 

- -0388 

— -000 2 



■3457 



-2-62IIJ 



•4871 



-I-682S 



-5'=424 



-4-cS.)ij 



86 



Mr. Ernest W. Brown, Theory of the 



B 



21. 


2t + C+m + 2g. 


2J + C+H1 — 2g. 


2i + c — m + 2g. 


2i + C — m — 2g. 


2J + c + m. 


2J + c-m. 




7 
5 








— '0001 








-•0031 




+ 'OOO9 


— '0010 


+ -O0IO 


\ 




— ■0083 


-•1471 


+ -0045 


+ "1262 


- - °943 


+ •0747 




i 


-•5658 


— -1190 


+ -0279 


+ '0681 


— 2-9362 


+ •9469 




—i 


+ +864 


-•3057 


-1-5007 


+ I-8829 


- T3 2 7 


-•3019 




-3 


+ '0646 


+ •0295 


- '1013 


- -1163 


+ 1-1756 


-■8338 




-s 


-•0059 


+ •0011 


- -0178 


— '0017 


+ -0249 


— •0238 


; 


-7 

Sum . . . 


+ '0005 




— -0003 




+ -0004 


— •0001 


1 

1 


-•0285 


-•5443 


-1-5877 


+ 1-9600 


-1-9633 


— •1370 



w,,v.,£~ l -7-aee'k 2 a 



2/. 


21 + c + m + 2g. 


2 i — C — III — 2g. 


2i + c-i-m~2g. 2('-c — m + 2g. 


2( + C-m + 2g. 


21 — c + m-2g. 






— -ooi 








+ -ooi 


5 




-•088 


+ •002 






+ -037 






—•320 


+ •074 


+ '00 2 




+ -212 ! 


1 


+ •004 


■-•142 


— •21 


+ "014 


+ •005 


+ 1-054 


_ T 


— -012 


+ •392 


+ ■268 


+ i'33 


-•303 


— -029 


~ 3 


+ •622 


+ ■004 


-•047 


+ -056 


-•193 


— 'OOI 


— 5 


-•051 




— 'OOI 


+ -ooi 


+ •016 




-7 '. 


— -ooi 












t Sum... 


+•562 


-■155 


+ ■086 


+ 1-403 


-•475 


+ 1-274 



] 

21. 


2J + c — m — 2g. 


2J — e + m + 2g. 


2i + c + m. 


2i — e — m. 


2J + c — m. 


2i-c + m. 


7 
5 


— '005 




— 'OOI 


+ '013 


+ 'OOI 


— '009 




— -211 


— -005 


— -048 


+ -6n 


+ "034 


- -857 


1 


— I'04O 


- -687 


-3-090 


+ I-549 


+ -850 


-5 - 45 j 


-1 


-i'3i5 


+ I-I37 


- '698 


- -482 


+7-75 


- -326 j 


-3 


+ -084 


— -249 


- -536 


+ -027 


+ -005 


— '014 | 


-5 
-7 


+ -ooi 


— 'OOI 


— -005 




+ -007 




Sum... 


-2-486 


+ -195 


-4-378 


+ 1-718 


+8-647 


-6-656 



Motion of the Moon. 
163. Characteristic k 4 a. Values of 



«7 



21. 


2t + 4g. 


2i'+2g. 


,2?.. , '. 


7 " 








5 




+ '0002 


— "0003 i 


3 




+ '0064 


-•0497 ; 


1 . . • 


+ •0029 


+ •0228 


+ •2695 1 


—-I 


+ ■0658 


-•2685 




; -3 . 


+ ■0425 


— ■0693 




! -s 


+ •0025 


— - 000I 




-7 
Sum ... 


— - 0002 




'•■•"••-..".. 


+ 'II35 


-•3085 


+ •2195 1 



B 



7 


24 + 4S- 


2g. 


1 

2i. 

I 








S 




+ '0001 


— '0003 


1 3 




+ '0025 


- -0051 


J 1 


+ •0013 


+ - 2o66 


—4-7553 


•' ~~' 1 


— '0176 


— 2-6677 




1 -3 


+ •0765 


+ -2237 




i -s 


+ •0043 


+ -0006 




1 ~ 7 




— 'OOOI 




i Sum . . . 


+ "0645 


-2-2343 


-47607 



%.„£"" '-f-ak 4 a 



\ :: ' ; 
2«. 


28+4g. 


2*'-4g. 


28 f 2g. 


2i'-2g. 


2i. 


7 












5 




-•005 




+ '002 




3 




+ -049 




+ -139 


— '022 


1 . 




+ ■190 


— •006 


+ 728 


— 2'37I 


— 1 


+ "002 


+ •001 


— '921 


+ 715 


+ 5'93i 1 


-3 


—•005 




-•066 


+ "004 


- "015 


-5 


— •010 




— •004 






-7 












Sum 


— •013 


+ •235 


-•997 


+ 1-588 


+ 3'5 2 3 



Mr. Ernest W. Brown, Theory of the 
164. Characteristic k 5 . Values of 



165. Characteristic^^ 1 . Values of 



i 






A 






B : 


i. 


2i + 3g + 2e. 


2i+3g — 2C. 


2i+3g. 


21 + g + 2C, 


2»' + g — 20. 


2* + g. 1 

1 


; 4 ; 




— •00002 






— -OOOI3 


— •00001 


3 




— '00038 


— "OOOOI 


— '00002 


-■OO554 


— '00098 [ 


t 2 


+ '00002 


— •O1872 


— •00156 


— 'OOI02 


— '10446 


— •02841 1 


1 


+ •00132 


— '61256 


— '09289 


+ 'OOo8l 


+ •00561 


—•23460 1 


1 


+ -02SI5 


+ •00677 


— 3'348o7 


+ 1-66897 


+ '0333 T 


! +'01907 | 


1 

i _I 


-■45559 


— •O0667 


+ '04340 


— '00167 


+ •16980 


— -00656 


— 2 


+ •00526 


+ •00524 


— '00258 


— -O0123 


+ •00455 


+ •03469 


_ 3 


— '00002 


+ 'O0O07 


+ -00064 


+ "00041 


+ •000 1 2 


+ •00074 


-4 


— - oooo6 




— "00002 


— -OO004 




+ •00003 


Sum... 


— •42392 


— '62627 


— 3' 4 oio9 


+ I-6662I 


+ '10326 


| — '21603 







A 




\ 


B 




i. 


2 «' + SS- 
— -00004 


ai + 3g. 

+ •00004 






it + g. 




3 




+ "O0OO2 




2 


+ '00002 


— -00003 




+ •00027 




1 


•OOOOO 


— '00056 






-■01230 







+ "00036 


+ •00672 






-•01946 




— 1 


+ ■00345 


— '01322 






-"00932 




— 2 


+ '00221 


+ •00029 




+ '00224 




-3 


+ 'OOOOI 


— 'OOOOI 




+ •00009 


1 
% 


Sum 


+ •00601 


— '00677 




| 


-•03846 


! [ 

1 I 






•/ — is k .-f-ak 5 










i. 


2* + Sg- 


2i + 3g. 




2 J + g. 


3 










2 






+ 


OOOOI 






1 




— 'O0002 


— 


00147 




1 





+ •00001 


+ '00067 







gk-= " 


-■00883 j 


— 1 


+ ■00033 


-•03317 


+ 


•02556 






— 2 


+ •00258 


+ "0O02I 


+ 


■00035 




i 


-3 


— '00005 










■1 


Sum 


+ •00287 


— •O323I 


+ 


•02445 



imp 
Up 

■ 



Motion of the Moon. 



So 



v'-ICk.e.-HlkV 



1 



1 



fli 

I 



;:■■■ ■', ■; ' : ;■:■ 
'::'■■:. ■■ : ': : : 


2l+3g + 2C. 


2i + 3g-20. 


' *'+3g- 


2i + g+2C. 


2J + g- 2C 


2*>g. j 


4 
3 




— -00001 






— ■00025 


— "00002 I 


2 




— -00079 


— ■00003 


— '00002 


-•01397 


— '00117 1 


1 


+ '00003 


— •07184 


— ■00368 


+ '00012 


+ •00619 


— •O2814 





+ •00088 


+ •1018 


-•35528 


+•18055 


—5728 





— 1 


-•04391 


+ •01382 


+ '09779 


+ ■00192 


+ •01989 


+ •02079 


— 2 


+ '00605 


+ •00076 


+ •00223 


: + ■OB2O4 


+ •00019 


+ •00478 


-3 


— "00006 




+ •00011 


+ •00007 




+ '00004 '. 


-4 


— '00001 










:;\::K;:\--::;::y : :1 


Sum 


—■03702 


+ "0437 


-•25886 


+ •18468 


-•5607 


-00372 1 



+ •00875 



166. Characteristic k 3 ee' '. Values of 



P 

m 

1 

I 



H 



HI 



■ 



1 

J ■ ■ i, ':"■:' 


2t+3g + c + m. 


3i ' + 3g — 0— m. 


^i+2>g + o—m. 


2t' + 3g-c + m. 


\ 4 










3 




+ '0004 






1 2 




-•0038 


+ •0001 


+ ■0004 j 


1 1 


— •6001 ' 


-•1546 


. . ' • + - oo5 : 8 


+ •0232 | 


1 ° 


+ •0190 


+ •2873 


+ -0360 


-■4097 


i . — i- 


-■f'399 


+ •1250 


+ ■1364 


-•3864 


—2 


-■0138 


— •0023 


— -0023 


+ •0074 


1 -3 


— •0015 


+ -0003 


+ - ooo3 


■^•0002 


-4 


+ •0003 








Sum 


— ■6360 


+ ■2523 


+ •1763 


— 7653 



Mi 



ml 

:' 



II 

111 

m 



■■■"■' t.v 


2i + g + c + ta. 


2i + g-c— m. 


2i' + g + c— m. 


2i + g— c + m. 

: : ■ : : : : ;: : ■; ■ : y: ■/ ; : : : . ;y ;:y:j 


.4 




— •0002 




! 


3 


+ - 0002 


-•0050 


'. — '0004 


— -0002 


2 


+ '0026 


— -1081 


— "0099 


+ •0096 


1 


+ -0793 


-■5 6 i7 


— '1619 


-•3477 ' 


• 


+ i"335° 


+ '45 °5 ' 


-1-6058 


-•7741 


— 1 


- -1471 


-•0756 


- -4687 


+•3263 


— '•% '"'.' ■ 


+ -0033 


— '0012 


+ -0T73 


' +'0129 .;:' 


-3 


— '0012 


+ •0001 


+ 'oon 


•' ; + - QQ02 - :: • 


-4 


— -0007 








Sum ... 


+ I - 27I4 


-•3012 


—2*2283 


-•7730 



iii§ 



go 



Sum 



Mr. Ernest W. Brown, Theory of the 



t. 


2i + 3g + c + r 


■ 

3 




2 




1 







+ '0009 


: ~ l 


"•1385 


i —2 


+ ■0153 


«_- t 


— •OOIO 



•1233 



</=il 



-ak'ee' 



2t + 3g— c — m. 



2j + 3g + c-m. 



— 'OOOT 

— "0098 
+ •0880 

— toSo 

— -ooi 5 



-•0314 



+ ■0002 

+ '0O2 2 
+ •0353 

+ •0017 

+ •0003 



+ ■0396 



2J + 3g-c + m. 



+ •0013 
— "1026 

+ •3498 
+ '0064. 



+ •2549 



2 
1 
O 

— I 

— 2 

-3 



Sum 



2t + g + c + m. 


2i + g — c — m. 


2t + g + c — m. 


2» + g — o + m. 






— -O002 








+ •0001 


— •0078 


— ■0003 


+ '0005 




+ •0050 


—•2215 


— •0109 


-•1057 




I +-3491 


-•3813 


-•5112 


+ •6618 




i +'1272 


■ — '0260 


+•4053 


+•1364 




| +'0025 


— '0001 


+ •0088 


+ •0010 




— •0001 




+ •0001 






+•4838 


—•6369 


— ■1082 


+ •6940 





167. Characteristic W 2 . Values of 









A 






B 

1 


i. 


2t + 3g+2m. 


2i + 3g-2m. 


2t + 3g. 


2J + g + 2m, 


2J + g — 2m. 


1 

2S + g. j 

i 


3 




— *0002 






—•0005 


+ -0003 1 


2 


— "0004 


+ -0003 


— •0001 


— - 000I 


— '0242 


+ •0091 I 


1 


+ •0002 


+ •0082 


+ -0004 


— •0023 


-•3466 


+ -J547 | 





+ •0244 


+ •0374 


+ •0082 


+ •0646 


+ •0539 


-•0195 


— 1 


-•2550 


+ •0022 


+ •0993 


-•0734 


■0000 


+ •0083 


— 2 


+ •0279 


— •0014 


— •0180 


— -0109 


+ •0008 


+ •0066 





— •001 1 


— '0002 


+ - ooo8 


— •0010 




+ '0004 


Sum... 

i 


— '2040 


+ •0463 


+ '0906 


—•0231 


-•3166 


+ •1599 



Motion of the Moon. 



m 



i. 


2i + 3g + 2m. 


2l' + 3g-2m. 


2/ + 3g. 


28 + g + 2m. 


28 + g — 2m. 


2» + g. 




3 

2 


y y-: : -:"v: r ^-: : / ::y: : v ■ y 








— •0011 


+ "0004 




I 




+ -0003 






-■0473 


+ •0186 





— I 


+ "O02O 

-•3075 


+ -0047 

+ •13 


+ •0012 

+ •250 


+ ■375 
+ •117 


-•l68 
+ ■11 



— •026 


glcV 

= — •0090 


'•'• """? 


—•0286 


— "0006 


+ •024 


-■0015 


+ "0003 


+ '0009 




-3 


— -0003 




+ "0002 










Sum... 


-•3344 


+ •13 


+ •275 


+ •290 


— •II 


—■006 





168. Characteristic ke 4 . Values of 





1 




A 


B 


i. 


j 2«' + g + 4C. 


2i + g— 40. 


2i + g + 2e. 2i + g~2a, 


2i + g. 


4 


1 


+ -00086 


+ •00089 


+ •00018 


3 


+ "00001 


+ •00139 


+ ■00066 +'01631 


+ -00642 


2 


+ '00084 


— •OOOlS 


+ •02200 +-00715 


+ •12224 


1 


+ •02860 


+ •00114 


+ •42.113 ^'00042 


— •00718 





1 +-55401 


— •O783O 


— •06698 +"00067 


+ ■00061 


—1 


-•14499 


— •OI869 


— •00019 —'03848 


+ -00043 


— 2 


1 + -00098 


— '00088 


+ •00010 — ' OI 347 


— •00719 


-3 


1 + -00003 


— •00003 


— '00083 — '00065 


—•00376 


-4 


— "00006 




— '00045 — "00002 


— •00018 j 


Sum... 


+ •43942 


-■O9469 


+ '37544 —'02802 


+ •11157 j 








"/ — is ke «-^ake 4 




t. 


2! + g + 4C 


2! + g— 4c. 


2i + g+2C. 2i+g-2Q. 2i + g. 




4 




+ "00004 


+ - O0002 




3 




+ •00021 


+ '00001 +-00070 +'00013 




2 


+ •0000.1 


+ -00024 


+ '00044 + "00096 + "00495 




1 


+ -00054 


+ •00389 


+ '0l6lO +'00193 —"00085 




0:. . 


+ •02001 


— -00863 


— ■OO723 — '0121 .0 


= + '00027 


— 1 


— -01421 


— -00073 


— •OO085 —■OO473 —-OOI17 




— 2 


+ •00114 


— '00002 


— '00015 — '00055 —"00099 




—3 


—•00005 




— •00013 — -00001 —-00016 




-4 


— 'OOOOI 




—•00002 




Sum... 


+ ■00743 


— •00500 


+ •00817 —'0138 +'00191 





Royal Astros. Soc, Vol. LVIL 



9 2 



Mr. Erxest ~\X. Brown, Theory of the 
169. Characteristic keV. Values of 



i. 


&i + g + 3« + m. 


1 4 




1 3 




1 2 


— •0018 


J 1 


— '0464 


1 


— •6708 


| — 1 


-•2353 


1 ~ 2 


+ •0005 


1 -3 


— '0014 


1 —4 

1 


— •0015 


1 Sum 

1 


-'9S 6 7 



2i + g-3o- 


+ ■0027 


+ •0418 


+ •0191 


— ■0314 


+ ■0736 


+ ■0479 


+ ■0023 



+ •1560 



22 + g+e- 



2J + g — C — I 



2/' + g + c-m. 



i 4 




1 3 


— '00.10 


1 2 


—•0285 


i I 


-•4227 


1 ° 


— '2906 


— 1 


+ ■0182 


— 2 


— '01 1 1 


l " 3 


-•0187 


| -4 


— •0010 


j Sum 


-7554 



+ ■0011 

+•0338 
+ •4429 
+■0554 
—•0390 
+ ■0263 
+ ■0275 

+ ■0011 



+ - 0002 
+ -0052 

+ 'I33 1 
+ 1-3679 
+ -0476 
- -0529 
+ "0025 
+ -0054 

+ - 000I 



+ •5491 



+ 1-5091 



n/- 



!»» 



-akeV 



i. \ 


2* + g + 3c + m. 


2 j + g_ 3 C_ m , 




1 4 




+ •0001 




1 ■ 3 ■ 




+ •0032 




! 2 




+ •0096 




1 


— -0012 


+ •0274 




1 


-■0373 


+ ■0200 




— 1 


-•0525 


+ ■0029 




— 2 


+ •0002 


+ •0001 




-3 


— •OOIO 






-4 


— •0001 






Sum 


-•0919 


+•0633 





21 + 2 + 3c-m. 



+ •0001 

+ •0044 
+•0485 
- -0205 

— '0019 



+ -0306 



2j + g + 3c-m. 


2s + g-3e + m. 




— "0002 


+ '0002 


— •0057 


+ •0074 


+ •0059 


+ •1638 


— - oo6i 


+ •8067 


-•0580 


-•0775 


-•0865 


+ •0025 


— "0071 


0000 


— '0002 


+ -0003 




+ -9034 


-'1579 



2! + g — o + m. 



•0057 
•0866 
•0704 

"°°55 
•0633 

••0729 

■•0051 



—•3097 



2» + g-30 + m. 



— ■0004 
+ •0023 
+ •0054 

— '0192 
-■0057 

— •0002 



-■or7S 











Motion of the Moon. 




y.? 










«/ — is keV -f-ake 3 e' {continued) 








i. 


2i + g + c + m. 




2i + g— e— m. 28 + g + c — m. 


2i + g— c + m. 


1 




4 










1 




3 






+ '0010 +'0001 


— 'O0O2 






2 


— "ooo8 




+ ■0310 +'0038 


—•0056 






I 


-'0253 




+ •0220 + '0886 


— ■0219 









— '0760 




+ ■0329 +"0157 


+■■005:2 






— I 


-■0152 




+ '0086 + "0450 


-•0258 


1 




— 2 


-•0059 




+ "00l8 +'0012 


—■0052 






— 3 


-•0015 




+ '0004 


— '0002 






-4 














Sum ... 


-""47 




+ •0973 +T548 


-•0537 


1 








I70. 


Characteristic keY 2 . Values of 












A 








■;'.*• 


28 + g + 2C + 2m. 




2S + g — 20 — 2m. 2i + g + 2C— 2m. 


2i + g — 2C + 2D1. 




■■/ 


4 






+ •0026 








3 






+ •0576 + -0007 


— •OO39 




■:■■ 


2 


+ ■0007 




+ •5520 + "0296 


-'I33I 




v/ v 


I 


+ '0020 




—•0432 + -4848 


- '0045 




■/ : 





—2683 




+•0063 + -7659 


+ •0171 




■■■:■: 


"I 


-T694 




+ •0093 — '0082 


— '2014 






-2 


+ "0240 




—•0005 — -0057 


—•0288 






-3 


— •0304 




+ -0036 


— •0014 






—4 


— •0024 












Sum ... 


-•4438 




+•5841 +1-2707 


- -3560 





:!■:. ■'■.■:."■:: 

1 i. 






A 




B 


2t" + g+iC. 


2! + g — 2C. 


2i + g + 2m. 


2i + g— 2m, 


2»' + g. 


1 ' 

4 




— '0008 




+ '0003 


— '0002 


3 


— '0003 


-•°i57 




+ '0172 


-•OO57 | 


2 


— '0109 


—•1408 


— '0191 


+ '358S 


— H83 


• r 


—•1984 


—•0073 


— •7306 


+ 2-1857 


-•7709 j 


... •. 


-•2647 


— •0081 


-■0518 


— - 0337 


+ "0233 | 


1: 


— '0102 


+ •1564 


+ •0428 


— "0030 


— 'on8 I 


\ —2 


+ '0010 


+ •0163 


-■1725 


+ "0199 


+ '1020 j 


: -3 


+ ■0139 


+ '0006 


— •0174 


+ '0005 J 


+ '008l 1 


: ~ 4 


+ -0009 




— '0006 






Sum...; 


-•4687 


+ '0006 


— ■9492 


+ 2 - 5457 j 


1 
-7735 | 



94 



Mr. Eenest W. Beowk, Iheory of the 



V-is te v»-r-akeV 2 




28 + g+2C + 2m. 



2l-t-2 — 2C — 2m. 



2i + g + 2o — 2m. 



I. 



4 
3 

2 

I 

o 

— I 

— 2 

-3 

—4 



Sum. 



4--OOOI 

— '0264 

-•2254 
—•0252 
-•0054 

— •0001 



+ •0001 
+ •0027 
+ -0844 

+ •0735 
+ •024 

+ •0011 



+ - ooo6 
+ •0198 
+ "°934 
+ •17 
+ •011 
+ '0005 



-■2824 



+ •186 



+ •30 



2! + g + 20. 



2i + g-2C. 



24 + g + 2m. 



2i + g-2m. 



— '0002 

— ■OO76 

— •0287 

— •032 

— •OOI 
+ '0021 



— •OOO7 

— •Ol88 
+ '020 

+ •13 

+ - OI93 
+ - 0007 



— •067 



+ -iS 



•316 



+ '5o 



171. Characteristic kee n . Values of 



2S + g-2C + 2m. 



— '0002 
-•OI59 

-•13 

— *040 

— •O280 

— •0013 



21 + g. 





+ '0004 


— 'OOOI 




0008 


+ -0IS7 


— "0048 




0782 

142 


+ •2968 
+ •089 


-•0925 




= + •0104 


067 


+ •10 


+•036 




•0271 


+ •0027 


+ •0140 




•0008 




+ '0004 





—•047 



i. i 


2t + g + C + 3m. 


2«' + g-c-3m. 


2i + g + c -3m. 


2i + g — e + 3m. 


4 ! 




+ -0004 






1 
3 1 




+ -0242 


+ '0024 


+ '0002 


2 




+ -4352 


+ -0703 


+ •0057 


1 


— 'OOI2 


+ I+45 1 


+ -8945 


— •0051 





- -i5 l6 


- -1946 


+ '3422 


+ •3416 


— 1 


+ i'3342 


— -0141 


— '0203 


-•2245 


— 2 


— -1292 


— -0006 


— •OI4I 


— •0671 


-3 


- -0256 




— "OOOS 


-•0039 


—4 


— '0014 








Sum ... 


+ ro252 


+ 1-6956 


+ I-2745 


+ •0469 



i 



Mnl',,,,1 ,./' //(, M,l,,i,. 









95 



illllllBIli 


t. 


2i + g + c + m, 


2s + g— c — m. 


2£ + g + c — m. 


2i + g— c + m. 




4 




— -oooi 






^^^^^li^lii 


3 




— '0102 


— •0013 






2 


+ -0020 


- T742 


-•0316 


— '0191 


I 


- "O07S 


— •6600 


-■3931 


+ '4424 


ppwv 





— /6609 


- -7S27 


+ 7341 


+ -8012 


liipslfiiltppls 


— I 


- -4825 


+ - oo6o 


+ •3980 


+ -2141 


■Rv 


— 2 


+ -H55 


— -0035 


+ •0141 


+ -0480 


^^^^l$|||l||::-:-:;;'!;:; 


-3 


+ "0152 


— -0002 


+ '0007 


+ "0022 




-4 


+ "0006 




— 'OOOI 






Sum... 


-~roi76 


-i - 5949 


+ •7208 


+ 1-4888 


|I«|ilSK«H»: i 












pit 

mm 












^^ftlllll 






v'-isw.- 


r-akee'- 1 




Hi' 


«'. 


2i+g+c + 3m. 


2i+g-o-3m. 


2S + g+C — 3m. 


2i + g— e + 3m. 


^Bill: • 


3 




+ -0008 






llpl^ 


2 




+ ■0341 


+ '0021 


+ -0003 


pit/ 


I 


— - 000I 


+ 7392 


+ •o'J34 


— -0023 


^^811111= 





— -0367 


+ •1646 


+•1380 


-•3056 


^^Klllil: 


— I 


— 1-3081 


— ■0031 


+•0159 


— •1227 




— 2 


- -i°53 




— '0044 


— •0054 


-3 


— -0023 






— -ooor 


^^|fSg:g:| 


Sum... 


-i - 45 2 5 


+ -9356 


+ •2150 


-'4358 



■pir 












|I|p8iKRtti:»?S;¥ 












llllt 

WBmffMMimm 


i. 


2t + g + e + m. 


2t' + g-c— m. 


2s' + g + c— m. 


2i + g— c + m. 


3 




— -0003 






^tajiiiii; 


2 


+ '0001 


— '0124 


— '0009 


: —■00 II 


^^^^^^^^0^y?/yM: 


1 


— ■0006 


-•2550 


-•0253 


+ •1352 


'- 





— T712 


+ •6440 


+ '2329 


. --6877 


^fclllll 


— i 


+ "43IS 


+ -0046 


-•3420 


+•0838 


^^Slfi ; ; 


— 2 


+ •0628 


— -0002 


+ -0039 


+ ' 00 35 




-3 


+ •0012 




+ •0001 


+ '0001 


lilt/ 


iNiiii... 


+ - 3 2 3S 


+ •3807 


- -I 3i3 


— '4662 





mt^^^lfS-^§^ 1 




Mr. Ernkst W. IIrown, Tlmn-ji of tl»: 
\~i. Clitirtt'ii'i-ixt'u'V'e.a. A alue* oi 






2i. 


24 + jb'-' '■ 


2i'--3--r-C. 2I + .i.'-r.:. 


2.+ ?-.'. 




^^^^^^^^^Iiiilljii|jli!|lii9^?ii|!l5s 




..... . _ 





Bi|Sll|il§Ssll|ilM 




7 I 






+ -oooS 






llSiltp 


+ -0002 


— TOOI + 'OO20 


+ '0364 


S|#|3S-; ; 1^B 


^^^^^^^^^^^iBIiililiiiliSlilil 


3 


+ '0015 


+ -0149 + "i5 86 


+ 1-1788 






SlililSiltll 


+ -0491 


4- >Si2 +4' 6 i T 7 


- -I4S7 


'tifelC villi 




— i 


+ 1-5291 


4- -0369 - ^077 
4- -0151 + '-57^ 


— -0321 
+ -i66S 


1 

1 


— j 


+ '0723 




-5 ! 


— -0063 


4- -ooSo + '04^2 


+ 006S 


;';yMS?"--1:^i 




-7 ' 
Sum... 


+ '0021 

4- 1 -6480 


4- -0003 + 'ooi 7 
4-1-0563 +4-6669 


+ -0002 


SiiipiliilSSfli^H 


+ I '2090 








v — 1.-. 1: Cj +-ak 3 e« 








'■ 2i. < 


2i t 3g + e. 


2i- + 3i;--c. ai + a-i-c- 


21 rg- 0. 





5 




3 ' 




iiiiiiili 


+ •0019 


— 1 


+ -I54S 


-3 


+ M2S 


-5 '. 


+ •0065 


-7 ; 


+ '0004 



+ - 0001 



i Sum. 



+ -292 



+ 


■0006 


+ 


I09S 


+ 


i7' 


— 


•°33 


+ 


•001 1 



+•250 



4. 


•0065 


+ 


•523° 


— I 


•264 


— 


•090 


+ 


•0059 


+ 


•0001 



•8(S 



+ -0016 

+ i5°7 

+ -966 

+ - °95 

+ -0217 

+ •0003 



+ I-235 



2(-i-3gim 



5 ; 


+ -0003 


3 


+ •0020 


1 


+ ■0645 


1 : 


+ -S9 2 3 


3 : 


-•7884 


5 ■ 


+ •0212 


7 ■ 


+ •0014 



173. Characterifitick z e'a. Values of 



Sum... 



•1067 



2i+ 3;r-m. 



— -0003 
+ -0032 
+ '0619 
-1-7294 

+ -35°5 

— -0150 

— '0007 

-1-3298 



21 + g + m. 

— '0003 

— -ooSS 

— -42S 6 

— 13-0974 

+ '95 > 5 

+ - 5°34 

+ -0257 

+ -0006 

— 12-0509 



+ -oooi 
+ -007s 

+ -3012 

+ i*5 6l 7 
-11-1432 

— -H49 

- -0047 

— -oooi 

- 9-3921 









Motion of the Moon. 


97 








sj — iz kv ,-~sk. s e'a 






21. 


2i + 3g + m. 


2« + 3g-m. 2i + g + m. 


2»' + g-m. 




7 

S 




— '0003 


-+- '0002 




3 


+ '0001 


+ ■0001 — -0292 


+ •0209 




i 


+ '0038 


+ •0034 —37306 


+ - 59 2 3 




— i 


+ '1406 


—•4881 — -7886 


+ 9'45 28 




-3 


+ 738S 


— -3000 + -2256 


+ •0016 




-5 


+ ■0192 


— •0088 + '0021 


— '0002 




-7 


+ •0001 


— •0001 






Sum 


+ •9023 


-7935 -4-32IO 


+ 10-0676 




1 74. Characteristic ke 3 a. Values of 










A 




Ill 


* 


2s* + g + 3e. 


2J + g — 30. 22 + g + C. 


! 
21 + g— C, 




9 




— 'OOO I 






7 




— -0089 — -0003 


— •0031 




S 


— 'O00I 


— -06ll — -0122 


— - I02O 




3 


— -0141 


+ ■0071 - '399° 


— -4204 




i 


-•5143 


— •0045: —-4881 


+ - 0II3 




— i 


+ ■6224 


+ •0981 +-0182 


— '0063 




-3 


+ •0150 


+ •0219 —-0105 


■f -0946 




-5 


— '0028 


+ •0006 +-0304 


+ - OI59 


! -7 


+ •0029 


+ '0039 


+ '0002 


■■ :■■■■ . 


-9 


+ '0004 


+ •0001 




Sum 


+ ■1094 


+ •053* --8S75 


— •4098 






■/ — is ke!>a -r-ake 3 a 




2t. 


2i + g + 3c. 


2t* + g— 3c. 2i + g + e. 


2i + g— e. 


7 




— •0004 


— 'OOO 1 


1 S 




— •0087 — '0003 


-•0043 


1 3 


— •0003 


-•015 --0157 


-'0533 


1 i 


— -Ol()l 


-•047 — -0554 


-'°73 




— i 


+ ■0641 


+ •0113 +-iii 


+ '034 




-3 


+ •031 


+ '0009 + -020 


+ •0123 




-5 


+ -004 


+ -0044 


+ -0007 




-' ! 


+ "0005 


+ - 0002 






Sum ... j 


+ •080 


—•059 +-064 


—•084 







9 s 



Mr. Khm:s-i W. Ukow.v. T/.-onj of t'» 



175. Characteristic ke 2 e' a. Values of 

A 



28. 


28 + g + 2C + m. 


2i + g — 2e— m. 


2i + g + 2c-m. 


22 + g — 2C+m. 


2i+g + m. 


2i + g— m. 


3 


9 ! 




— '0002 




+ - OOOI 








7 




- -0074 




+ -0076 


+ -0018 


— -ooio 




5 ! 


+ -0047 


— -2310 


— '0004 


+ -1880 


+ '°7'^4 


- -0432 


1 


1 
3 


+ "1891 


- -5328 


— -039 2 


- '4235 


+ 2-0144 


-1-0514 


'■§ 


1 


+ 47223 


+ '2560 


- -6lS4 


-3-1861 


+4-S725 


+ -I2II 


B 


— 1 


— 'i3 28 


- \3295 


— 27916 


+ -2076 


- -0405 


+5-5200 


4 


-3 


- -4499 


— -1708 


— -2700 


+ -1003 


+ '2134 


- '2235 


1 


-5 


+ "053* 


— -0058 


- -0183 


+ -0035 


+ -0769 


- -0745 


: j 


-7 


+ -0137 


— -oooi 


— '0080 




+ -0021 


— -0024 


1 


-9 


+ -0003 




— -0002 








a 


Sum... 


+ 4-4005 


— I'02l6 


-37431 


-3-1025 


+ 6-9190 


+4-2451 










n/ — IZ teV .-5 


-ake'Va 


2i + g + m. 


2i + g — m. 




2i. 


2t' + g + 2e + m. 


2S + g — 20-ni. 


2i+g + 2c— m. 


28 + g— 2c + m. 




7 




— - O002 




+ -O002 






m 


5 


+ •0001 


— •0170 




+ -0124 


+ -0022 


- -0013 




3 


+ '0050 


-•2344 


— •001 1 


- -1335 


+ -1255 


— -0708 


'■'I 


1 


+ , z7 I 5 


-•2173 


-•0391 


+ 2-7063 


+ 1-3020 


+ -0216 


1 


— 1 


— •0299 


-•O979 


— ■8122 


+ -0885 


+ -0199 


-4-6849 


•M 


-3 


+ -4ISS 


— •0I08 


+ •2425 


+ -0070 


+ -0978 


— -1008 


m 


-5 


+ •0347 


— •0002 


-•0068 


+ 'OOOI 


+ -0058 


— -0052 


m 


-7 


+ •0011 




— •0006 




+ -oooi 


— -oooi 




Sum... 


+ •6980 


-■5778 


-•6173 


+ 2-6810 


+ 1-5533 


-4-8415 


1 

■"''""^ 
°* H 

: " : " : H 

m 

:M 






























1 
















1 



'vmm 



WA 









I 










M 






J 



Motion of the Moon. 



99 



CHAPTER VIII. 

Terms of the Sixth Order. 

Section (i). Formula and Methods of Procedure for u. 

ij6. Terms in only two characteristics, e 4 k 2 , e 2 k 4 , have been calculated ; for those 
in e 6 , in which very small divisors do not occur, the elliptic values can be substituted ; 
those in k 6 are insensible. Xo terms with arguments 21 are calculated, as no small 
divisors are present ; the constant C enters only with these terms, and it may therefore 
be neglected. 

The method is that of Chapter VII.. with Q 1 = 1/ = A' = o. Also a takes the 
values 6, 5, 4, 3, and b = 6-a. For u 3 u 3 , s 3 s t , z 3 z 3 , Dz 3 J)z 3 , u 3 s 3 + s 3 u 3 = 2u 3 s 3 , &c, 
we must substitute the halves of these functions when dealing with the general 
formula). 

177. The main difference arises in the development of Sf. S(l)f ). Here 



*i 8 £4 , dii , , ,B 2 f9 , , ,3 2 f, , of, , of, 



00 og oV 



eg- cc eg 



with a similar expression for S(i)f ) ; o 2 f 2 /3c3g, o 2 (Z>f 2 )/8c3g arc zero. 

For the first two and last two terms the formula) of § 134 are available with 
a = 4, 3, 2 and b — 4— a, a=2, 1 and b=2—a ; when a = b the remark at the close of 
§ 176 must be noted. 

For the other two terms I find, for the coefficients of £ ±(2,+1 ' c > only, 



8 2 f, 



do' 






6c' 



?■■■£.. 



ox* d 2 

P^ = ( 8M l'" 8 + 8 k» M o) » g-^(-D f/ 2) = -(8M k ,S -%,M ). 



No general formula) for the derivatives of Q were obtained, owing to the difliculty 
of expressing them in convenient forms. The cases are 

a > b = 3, 1 j characteristics ke 2 , k ; k 3 , k ; 
a, b = 2, 2 , „ ke, ke ; 

a, = 1,1, „ k, k. 

Royal Astrox. Soc, Vol. LVIL p 



100 



.Mr. KkM.M W. !>:;.. w.v. Ti>- •r<j 



>f the 



For these cases we have, using the notation of § 134 for 3" and of § 135 for :', 

/i Q = 4 2'^2)( ! :. l = t )- 2 a'^. 1 2>,. 1:> 



M 

§1 



a = ke 2 or k 3 , 



: b = ke, 



I og J og 

dc 



2 S'™ s \ e )-2 5 ' ke Z? 5k0 ; 



: 6 = k, 



: ,-^-r =2S'^(5 S k )-2S' k 2)s k , 



dg 
;■■-■«: j 



Vdg/ 



3 2 Q 



where (^ ta ) K+c denotes that the terms whose arguments are ±( 2 i" + g + c) only are to be 

used, and similarly for (y k e) B ^c 

178. The calculations were made on the plan outlined in Section (iy.) of 
Chapter VIL, with certain abbreviations. As in most cases only three significant 
figures were necessary, it was found possible, in forming the products of series, for the 
computer to add each pair of logarithms from the slips and look out the number 
corresponding to their sum from a four-place table (printed on a card) without writing 
anything down (see § 140). Only this last number was actually written down, and 
thus the sheets on which the various series were added together could be entered 
straight from the multiplication slips. The great majority of the products consisted 
of numbers with one or two significant figures, and long practice has made us so 
familiar with the logarithms of numbers from 1 to 99 that a glance fit the tabic was 
rarely necessary for these. 

Nearly all the other operations were so arranged that the use of logarithms was 
not necessary and the computation sheets were much abbreviated. In the few cases 
where logarithms had to be used they were written down on a spare corner of the 
sheet, so as not to disturb the general plan. 

Only those sets of coefficients corresponding to the arguments 2z±(2c-2g) 
required the special method of § 138. 

Section (ii). The Homogeneous Equation for z. 

179. As the calculation of B t (see § 139) would have been long, the homogeneous 
equation (8) of § 7 was used. The terms calculated were those with characteristics 
k 5 e. k 3 c 3 . ke 5 , so that we have i2 t =o. The equation is then 



Put u'^utr 1 , *■'=&£, and divide by & The equation may be written 
f" + 8f"=(i) a — 1 - 2m- S -m 2 )«M'— 2(D + 1 +m)%Du' — hnhs'^—o, 



■ 



! 1 

ins 



(1.) 



Wk 

If 



Motion of the M< 



oon. 



IOI 



or 



fact, 



(2) 2 -g 2 )su'~ 2(I> + g )zDu' + (g ' z — i -2m— im 2 W+2(g -i— m)sZ)w'— ?m 2 ss'^ 2 =o. (2) 

2 2 

The constant coefficients in the last three terms of this equation are small. In 

g 2 — i — 2m— 5m 2 = — -00044, go— r— m = + "°043 2 . 3 m 2 = + -009805. ... (3) 

Also, since Jhi ' has the factor in 2 , the term of principal importance in the determina- 
tion of ~ 6 i*> the first. If then the operation (V.^ 2 — </„ a ) _1 he performed on the equation , 
the first approximation to z 6 is given by 



"~ -* + U-go B>- gl * )~ Du + 



i+2m+ _m 2 -g 2 



D*-g 



i>--go 2 2 



where wc neglect ~ G on the right. I: 1 or the second approximation we substitute 
— ~6 (1) ( M o' — a ) (§ J 3 2 ) f° r tne nrs t term on the right and c 6 0) , lht ' for c, //«' in the 
other terms ; and so on. 

180. As in Chapter VII., c , g are to be used for c, g in performing operations 
involving D. Let Sf" denote the parts of (1) due to c — c„, g — g (1 . Then 

8f"=c 2 + +g 2 -±- +c 4 -J- +g t -J- ; 

do dg do dg 

the terms involving second order derivatives are found to be negligible, owing to the 
smallness of c/, c 2 g 2 , g 2 2 . 

Let q denote an index of £ in v.'. Then, using the notations of §§ 134, 135, 

8 *"=2'^£ j 2D(zu')-2zDu'\ - 2(D+ 1 + m)s^'^u'. 
oc do l J oc 

Since 1+111— g is small enough to be neglected when multiplied by c — c„. g—go 
(there are no small divisors), this may be written : 



do do do 



with a similar expression for f}f"/dg. 

Hence, after putting c = c l): g = go hi the coefficients of £ in (4), we must add 

1 sf" 

■0 s -go 8 

to its right-hand member ; this expression separates into two parts, involving the 
operators (J) 2 — go' 2 ) -1 an d (-^~~&>)~ l respectively. 

181. The actual computations are comparatively short. The products 

z a u\, x a Du\, with a, b = 5, I ; 4, 2 ; 3, 3 ; 3, 1 ; 2, 2 ; I, I, 

are obtained according to the plan explained in § 140, and the other remarks there 
made apply here also. Moreover, as there are no monthly terms, small divisors are not 



102 



Mr. Ernest "W. Beown, Theory of the 



present. In most cases the terms multiplied by (3) can be altogether neglected ; when 
they are not quite insensible a simple inspection shows what additions arise from 
them. It was also found that a second approximation to the value of z 6 was not 

necessary. 



Section (iii). Values of A x /ar\, B^/aPX, ?/ A £ _1 / a \ t^/aX. 

182. The tables giving the characteristics and arguments calculated are 
(see § 148) : — 



§• 


\. 




Arguments. 


I 


183 

184 


e% 2 
e 2 k 4 


2l±4C±2g, 
2t±20±4g, 


27±4C, 2i±2C±2g, 22±2C, 
2l±2C±2g, 2?!±4g, 2l±2C, 


2l±2g ! 
2Z±2g | 












§• 


X. 




Arguments. 


1 


185 

186 
187 


lc'e 

k 3 e 3 
ke r > 


2i±5g±c, 

2i±3g ± 3 c > 

2i ± g±S c ! 


2i±3g±c, 2i±gdto 
2*±3g ±C . 2i±g±^c, 
2i±g±3 c > 2i±g±c 


2t±g±C 



The terms for which the method of § 138 was necessary, owing to small divisors, 



were 



Arguments. 
±(20 -2g), 



Characteristics. 
e% 2 , e% 4 . 



I" §§ l8 3> l8 4 the arrangement of §§ 149-163 is followed exactly. In §§ 185-187 
the final results for «^/a\ are alone given, as there was no definite stopping place in 
the computations, and the first approximations are the final results. 



i 



m 



i 
■ 



Motion of the Moon. 
183. Characteristic e 4 k s . Values of 



A:- 



J 03 



» 


2l' + 4C + 2g. 


21 + 4c - 2g. 


zi + 40. 


28 + 2e + 2g, 


21 + 2C — 2g. 


21 + 2C. 


21 + 2fT. 


3 










— "OOOg 






2 




— •002 


— •001 


+ "002 


— ■O233 




— 'OI4 


1 


+ •002 


— •045 


+ ■007 


— - 02I 


-■OI57 


+ •101 


-•371 





+ •005 


-■O36 


+ '354 


-707 


-•I346 


+ •052 


+ "°59 


— 1 


-•079 


+ •008 


— •052 


+ -I 33 


+ ■0089 


— "019 


-■015 


— 2 


+ "OIO 


— 'OOI 


— '004 


+ ■001 


-•0674 


— •003 


— "004 


-3 




— - 006 




— "002 


— ■OO25 


— ■008 


-•O05 


Sum... 


— '062 


-•082 


+ ■304 


-'594 


-'2355 


+ •123 


-'35° 



B 



lijstllStt//. : 


8. 


1 21 f 40 + 2g. 


2i+4e— 2g 


21 + 4c 


2i + 2e + 2'g. 


28 + 20 — 2g. 


2! + 20 


2t + 2g. 


1 


3 










+ "0002 






Ililtliii®-* 


2' 










+ •0072 


+ •001 


— ■004 


1 ; 


I 




+ '012 


— "OOI 


— "020 


-■°493 


-•066 


+ •084 


1; 


O 


— •031 


-•°73 


—•179 


+ •142 


— ■O054 


+ ■104 


-•134 


I 


— I 


+ '027 


'000 


+•043 


-■058 


— '0037 


— '010 


+ •003 


BfcBlvi',' 


— 2 


— •007 


+ •003 


+ - 002 


+ •003 


+ '0196 


+ •004 


— •O08 


WSll0iMM> . 


-3 
Sum... 






+ '001 


— •002 


— •OO05 


+ •004 


+ "002 


||pt8;SM¥:»::"'' ■■ 

Illiiiiwiv'. •' 


—'on 


-•058 


-•134 


+ •065 


-■0319 


+•037 


-•057 









u e t V £ J -f-ae 


*k 2 






'if 


2i + 4Q+2g. 


2i— 4e— 2g. 


2j+4c— zg. 


.'2t-*4e+zg. 


22 + 4c. 


'.21 — 4c. 


3 








— ■001 




— "OOI 


■2 




+ •002 




+ ■003 




— 'OI 


:-I . 




— r oo6 


— '00 3 


■ . •. — '01 


+ •001 


— •001 : 


O 


+ •010 


+ '011 


-•044 


+•031 


+ ■059 


+ ■027 


1 -. 


-•028 


+ "001 


+ '02 


—'007 


"026 


— "OOI 


— 2 


+ •010 




—•004 




+ 'OI 




-3 


•00 








— "002 




1111 ... 


— "or 


+•008 


—•03 


+ •02 


+ •04 


+ •01 



■ 



104 



Mr. KliM.ST \Y. IJkow.v. rin,.,;i -\ lb, 
w . k ,£ ^-f-ae'k 2 (continued) 





r 












:•: : :-KvMvM ; : y>--:-y/x >■ 




m 


l. 


J 2! + 2C + 2g. 


2» — 20 — ag. 


2J+20 — 2g. 


2S-2C-f2g. 


21 + 20. 


2t-2C. 


2* + 2g. 


28 — 2g. 


"ik 


3 


1 


—•ooi 








-•003 




+ 'OOI 




2 


+ '001 


— 'OI 


— - O02 


-■015 


—•001 


— •OO4 




— "O06 




I 


1 + -oo6 

i 


+ '024 


-•025 


+ •032 


+ '022 


+ -OI 


- -078 


-•06 


■ ■■.■■.\:«.;j 


O 


— •145 


— '022 


+ ''4 


-•58 


"-•056 


+ "I02 


+ •086 


+ •064 


;| 


— I 


+ •083 


+ •007 


+ •003 


,+ ' 01 7 


•OO 


+ 'OOI 


+ :o3 


— ■Ol6 




— 2 


'OO 


H-'OOI 


— •OO4 


—•004 


— '002 




+ •004 


+ •003 


:J§ 


-3 






4- 'ooi 




— •OO4 




— •002 




J 


Sum... 


-•06 


•00 


+■61 


-"55 


— •04 


+ •11 


+ ■04 


— •01 


J 



1 84. Characteristic e 2 k\ A r alues of 

. A ,. . . 



i. 


j 2i + 2e + 4g. 

^:7:;:7;:;7::;:;x;;:::;:;::;;:;:;;V:7; 


2J+2C — 4g. 


28 + 2C + 2g. 


2J + 2C-2g. 


2% + 4g. 


21 + 20. 


2l'+2g. 


3 


! 






+ '0001 








2 




+ •003 




- "0051 


+ 'O0I 


— -ooi 


- '003 


1. 


IS- ••//• \S6D.; 


+ •178 


+ "002 


- -3548 


— '004 


+ 'OIO 


- -058 





fe^: : :- : Mr^ 


+ -8IO 


+ •039 


— S -I 477 


-•063 


+ I '696 


— 5-159 


— 1 


— •002 


+ •003 


-•I38 


- - 5573 


+ ■299 


+ -506 


— 1-219 


— 2' 


, +'017 


— - 002 


+ - OIO 


- -0056 


+ •027 


— -oio 


:■ 4r '-:00 : '% 


-3 






— •002 


— '0001 


— •001 


— 'OOI 




Sum.. 


+ •015 


+ •992 


-•089 


—6-0705 


+■259 


+ 2'200 


-6-478 



2 
I 

— 2 
-3 



II 



2J + 20 + 4g. 2S + 20— 4g. 2J + 2C + 2g. 2J + 2C — 2g, 



— -001 

— -oio 



Sum...! +-045 



2 j + 4g. 



+ •436 



— -406 



■■1399 



+ •283 



+ •006 




— •0028 




— - 00I 


+ •002 


— ■008 


-•1373 


+ •004 


— "014 


+ ■369 


-•665 


+•0896 


+ •267 


— i'66i 


+•058 


4- -268 


-■0743 


+ ■013 


— -148 


+ •001 


— •001 


—•0148 
— •0003 


— •001 


+ '027 

— - 002 



-'■;w 



2J + 2g. j 



003 

-<5 

062 

004 

oo:: 
-■■/).; 



' + 



Mil 
saHT 



Motion of the Moon. 



105 



• 


2J + 2C + 4g. 


2i'-2C-4g. 


2/ + 2C — 4g. 


2t-2C + 4g, 


2t + 2C + 2g. 


2! — 20 — 2g. 


3 






*00 








— ■OOI 


. 2 






+ •005 


+ ■001 






•OO 


1 






— ■042 


— •02 


+ •002 


+ - O0I 


— •O29 









+ 'OOI 


—•030 


+ '334 


+ ■003 


+ •272 


— 1 








-•073 


-■08 


— •O36 


+ *002 


— 2 


+ 


•010 




— ■001 


— - OOI 


— 'OI 




-3 




•00 








+ '0O2 




Sum... 


+ 


•01 


-•04 


— •12 


+•25 


-•04 


+ - 24 



('. 


2i + 2e — 2g. 


2i — 2C + 2g. 


2» + 4g. 


2t-4g. 


2'i + 2C. 


2i — 2C. 


2 S + 2g. 


21 — 2g. 


3 
















— "OOI 


2 


— •002 : 


— -002 




•OO 




— 'OOI 




— '029 


1 


— -182 


— '2IO 


—•001 


+ -065 


;." + '661 


+ -o8 


— -OI5 


~-i6 





- -98 


+ 2-SS 


— - 002 


-'353 


+ •443 


+ 1-725 


-1-818 


— •226 


— 1 


- -117 


- -o S 8 


+ - I44 


-■005 


-•19 


— '009 


+ 78 


— '002 


— 2 


- '013 


- -003 


— "02 




— ■006 




— '021 


— •ooi 


-3 

Sum... 










+ "OOI 




+ - OOI 




— 1-29 


+ 2-28 


+ •12 


-■29 


+ •25 


+ 179 


— I"07 


-•42 



io6 



Mr. Eknest W. Brown, Theory of the 



185. Characteristic k 5 e. Value of y/—iz Ve -r-ak b e. 



i. 


2i + 5g + e. 


2i+Sg-c. 


2t + 3g+0. 


22+35-0. 


28+g + C. 


•At+g—c. 


2 
I 








— *0I 




— 'ii 


O 




— "01 




-2 '45 


+ ■82 


-9-81 


— I 




+ ■08 


-■07 


+ -6i 


+ T4 


+ "24 


— 2 


+ •01 


+ •01 


— ■01 






' .""-' " 0I .' 


Sum... 


+ •01 


+•08 


-■08 


-1 -85 


+ ■96 


-9-69 



m 
m 



m 



186. Characteristic k 3 e 3 . Value of ^/— i2- kV -r-ak 3 e 3 , 



i. 


2i + 3g + 3=- 


2i + 3S-3e- 


2 j + 3g + c. 


2i'+3g-C. 


2/ + g+3C. 


28 + g-3C. 


2J + g + C. 


2»" + g -0. 


1 

3 












— ■01 






2 




— "01 








-■03 






1 




+ •01 




— •10 




+ •09 




— •02 







+ ■06 


—25 


+ •09 


+ •12 


-•l6 


-•03 


-•19 


—1 


-•°3 


+ •01 


+ •10 


+ •03 


-■°3 


+ •02 


+ "OI 


-•04 


—2 






+ *02 








+ •01 




-3 


















Sum... 


-■03 


+ •07 


-•13 


+ '02 


+ ■09 


-•09 


— •01 


-•25 



187. Characteristic ke 5 . Value of xZ—iz^-i-ake*. 



i. 


2i + g+Sc. 


2i + g-Sc, 


2« + g+3c 


2i + g-3c 


2i + g + c. 


2» + g — C. : 


3 








+ ■001 




+ 'OOI 


2 






+ •001 


— ■001 


+ ■004 


•000 


1 




+ '002 


+ •010 


+ ■003 


— •002 


•Aod 





+ ■010 


— •004 


— ■007 


— •002 


— •001 


—■005 


'• . — I*' 


— -009 

+ •002 


— ■001 


+ •001 

1 a 
+ -004 


— •004 


+ ■002 

—•001 


— •001: 


; -3 














Sum... 


+ •003 


-■003 


+ "009 


— •003 


+ •002 


— - 006 



111 

m 

m 
m 

111 
sill 

n 

■ 
i 

ill 
I 

sill 
■ 

Sill 
1 



iH 



Motion of the Moon. 



107 



CHAPTER IX. 

Results in Polae Coordinates. 
Section (i). Foil aula for Transformation. 

188. Longitude.— The formula are given in § 41. The value of V is obtained by 
special values from equation (48) and the other Y^ from equation (47) in that section. 
By development, using the notation of § 139. we obtain 

2lVj = («i-8,)-4-(m , s ), 

2l V 2 = (ws-Sj-iV + la, 2 )-^, s ), 

2.V, = (M-M)-^(« , So ), 

2lV 5 = (M 5 -S s -?' 2 M 3 + g 2 S 3 -?i 1 M' + S I M')-r-(M , S ), 

2 * V 6={M6-««-(«|«« + tt,«4+K*) + («l«« + V4 + W)+«l , M'-«, , M' + «»N-g,N}-l.(» 0> B ), 

where M J .(»„) = ^_!« , i''i+*j , i «iS L V 

■ { 0,; «„ 2 ~v~ + i^-^ 
M'-=-(« 0) ) = M-f-Ko-r^+VrA!. Nh-KO = -^-+ < M- M !S 

All the products and most of the sums in Y„ V 2 , Y 3 , V«, M, M', N, were at hand. 
For V, the product u 2 u 3 was available ; the only products to be formed were, therefore, 
«j ?h by j/V and M'~(m 0; ) by V^o- In Y,. the known factor ?/, m b + «, m 4 + 1h 3 s was 
multiplied by i/< Ml 2 /V by M'-=-(« .) and u 2 /u, by N-=- («„,). In all cases the 
corresponding functions of ,<■■ were obtained by putting i/£ for £ in w, 

189. Parallax.— Yot convenience in obtaining the latitude, equation (49), § 42, for 
the parallax was computed in the form 

">.= ! + (/•») +(p°)+(e?) +(p?) 

r \ r ' 1 v r ' 2 ^ r ' 3 > r ' .] 

for orders up to the fourth inclusive. Here 
(£?), = -l(«i+»i)-K«oi»o). 

(7), sss {-K«»+«*)+K«i*+*i s )+i» 1 «i-i2 1 ! }-f-(Mo,«o, Po), 



-H w o> So. Po). 

s 1-3 1*2 



- 1 " *-IS(«i*«, +«,»«,) 



-T 3 «(2ttl«^i+2« 1 » 2 M,+« 1 S» J +«,^) + | Sl *(« 1 + g s ) + 3 Sl - J („ 1+8l ) +1 ^„ i 4 + 

All the products and, by suitable rearrangements which differed with different 

characteristics, many of the sums were at hand. 
Royal A.stkon. Sue. V<>;.,. LY.1I, 

*4 



U>- 



?tll\ Ml£\i:sT W. Hli'iWX. 7'//"./ 1 // <;/' //'(" 



For the terms of the fifth order, in which the characteristics e\ e 3 k 2 , ek 4 were alone 
needed, it was shorter to use the Jacobian integral, equation (5), § 7. Here ^=0 
and C is not present in odd order terms, so that 

All the products had been obtained in finding ?/,. An examination of the errors 
produced by using c , g for c, g showed that they were insensible. The terms factored 
by m 2 are also insensible in most cases. 

The order of accuracy for the parallax is found by dividing the characteristics 
required in the latitude by k. 

190. Latitude.— To the sixth order inclusive this is given by (§ 43) 

p ft ft" r p° p 

J y ( V5i "R \ 

i=l x i=l Po ' 

where 
R, = (£») , R 2 = (J») o +1'^, R 3 = (£°) 3 + e«iS 2 -i~i 2 K+*i)}-^( w o. «o. Po). 

PO W-5 \ r ' 5 \° '5 

= (') +A.W^-(^) 9 }.+- • s - L !{i a «^-A?!(5 I +?)}- 

\r/5 Po Po L Po Po vw o s o / J 

The expression for (,cr/r 3 ) 5 in R„/p is obtained by multiplying equation (4), § 6, 
by z, putting il x = o and neglecting the terms factored by nr ; we can also put 
c = c , g = go in this expression. 

All the products in the B,_; were known except those hi the last term of T? 5 /p . 
When the R,-_, have been obtained the multiplications by z,/p a were straightforward. 



Section (ii). Change of the Arbitrary Constants. 

191. The usual constants c, 7, a used in the lunar theory are so defined that the 
coefficient of the principal elliptic term in longitude is 



that of the principal term in latitude, 

27— 2ye 2 — Jy s + -3%7 c4 > 

and a is defined by the equation 

a 3 n i = JE+M. 

The results of the transformation to polar coordinates furnish the following 
coefficients for these two terms : — 

+ •99972 871 e + -oi4S7 76 e 3 --o243see' 2 + 1-00499 9 ek 2 --oi7i ea 2 
+ •0021 e 5 — -046 eV 2 +'37i4 e 3 k 2 + -23 ee'% 2 +i'SS 3 <&*' 



Motion of the Moon. 



109 



and 



+ 1-99974 473 ^ + -99167 5k 3 — -25024 2 ke 2 +-ooo73 k«' 2 — "0098 ka* 
+ 1-48 k 6 — -795 k 3 e 2 + -o k%' 2 --0287 ke 4 --os keV 2 ; 
also (§ 44) 

a = "99909 31420 a. 

192. Equating these, I find 

e= + 2-00054 273 e — -36681 52 e 3 + -o4873 ee 12 — 2-01160 2 ey 2 + -o342 ej — ) 

+ ■049 e 5 + "3S eh' 2 — -246 e 3 y 2 — -56 ee' 2 y 2 + -9ii ey 4 , 

k= 4-i - oooi2 765 y— -49609 1 y 3 — -49924 3»/e 2 — -00037 ye /2 + -oo49 7 ("7) 

—■128 y 5 +ro7 y 3 e 2 + -o yV 2 — -095 ye 4 + -]2 yeV 2 , 

a= +-99909 314 a, -= +1-00090 768-. 

a a 



Section (iii). Numerical Values of the Constants. 

193. The following are the values of the constants used in reducing the results to 
seconds of arc :- — 

m=i73 25594"-o6, »'=i2 95977"'4i5 , m =-08084 89338, 
e = -05490 056 , e' = -01677 191 , y = -04488 716 , 



34i9 //, 59 6 > 



^ = 8"-78oo. 



E 
M 



81-500 . 



a E—M 

ll = a'-J+l/ = ' 002SOS32 



The value of c corresponds to «i coefficient 2 2639' /- 58o of the principal elliptic 
term in longitude : that of y to a coefficient 1 8461 "-480 of the principal term in 
latitude ; that of a to the value 342 1"- 700 for the constant term in the sine of the 
equatorial horizontal parallax of the Moon. 



Section (iv). Numerical Values of the Parts of the Arguments and Coefficients arising 

from the Various Characteristics. 

194. The coefficient of each periodic term in longitude and latitude is of the form 
XP (<r, c' u , y~, af). where X is the characteristic of the principal part and the factor of X 
is a quadruple power series proceeding according to powers of e 2 , d 2 , y 2 , af, with 
numerical coefficients ; in the sine of the parallax each coefficient is of the same form 
with the additional factor i/a. In the longitude and parallax only even powers, and 
in the latitude only odd powers, of y are present. In this section will be given the 
coefficients of all those periodic terms for which P is not limited by the calculations to 
a single term. The part due to each characteristic in. a coefficient of a given periodic 
term is separately shown. Those periodic terms which have had the parts due to the 
principal characteristics alone calculated are not set down in this section, since these 
parts are the final values for these coefficients given in the next section. 

The characteristics in all cases are composed of the new constants e, c, y, a u that 



m 



: lO 



Ml-. tliM.ST \Y. ili.-iWS. Tlf.r-' <; !/■■ 



is to say, they are the same as those of Delauxay, with the exception ai for a. A 
direct comparison of each part with the corresponding part given by Delauxay is thus 
possible, when allowance has been made for the slight difference in the numerical 
values for the constants used by Delaunay and myself, and for the change from a to a 2 . 
195. The following table gives the various parts of the annual mean motions of 
the perigee and node * due to the separate characteristics set down in the first column. 

Annual Mean Motions. 



Char. 


Perigee, 


': Node. 


1 


+ 148524-92 


1 

i —69287-90 




e 2 


- 5 r[ r3' 


— 616-09 




e n 


+ 156-27 


- 25-46 




■■■■.»"■; 


- 1739-85 


+ 260-59 




«i* 


+ 2-24 


I IT 




e 4 


+ -o 4 


+ '• "07 




e 2 c' 2 


- '99 


- '57 




eV 


+ 672 I 


■*- 1-70 




e' 2 y 2 


— i'6i ! 


+ -08 




r 4 1 


■- I'S 1 


+ -05 




Sum... ! 


+ 146426-92 ! 


—69672-04 





196. In §§ 197-262 are given the coefficients of those periodic terms in longitude, 
latitude, and parallax which contain more than one characteristic, The arrangement 
is primarily according to the orders of the principal characteristics, each principal 
characteristic being attached to a definite set of arguments which differ only by 
multiples of 2D and in the signs of the multiples of I, /', F. 

The notation for the arguments is that of Delaunay (§ 10). 

In the first column of each Table is placed the characteristic, and in the suc- 
ceeding columns the coefficients corresponding to the multiple (i) of 2D placed in the 
first row of each Table. 

In the last row is given the sum for each column, and therefore the final value for 
the coefficient of each argument. 

The coefficients in longitude are given in §§ 197-225, in units of o"-ooi. 

The coefficients in latitude are given in §§ 226-248, in units of o"-ooi. 

The coefficients in parallax are given in §§ 249-262, in units of o"-oooi. 

All coefficients have been calculated so as to be correct to the last figure given 
with the exception of those depending on e 6 , for which the elliptic values have & been 
substituted. This remark applies also to the Tables in section (v). 

* I have obtained the complete theoretical values of these two quantities and compared them with 
their observational values in the first of the two papers referred to in § 147, 



m 



m 



in 

111 
iH 

mm 

■A-mm 



$■ 

m 

#111 

Hi 
11 



m 

win 

m 

fill 

Sis 
|jj 

ill 

mm 

mm 



111 

m 

H 
im 






llill 

♦1II1 
Sill 

fill 
'mm 
mm 



mm 

■ 
till 

111 
'mmm 

m 



■ 



■ 



Motion of the Moon, 



in 



197. Arg. 22 - D. 



.... 






i. 


1 ' 
1 3 

1 . -. 


2 


i 


I 


; + 49 


+ 8740 


+2106246 


e 2 


.,—$- I -|- 67 


+ 5217 


+ 298973 


e' 2 




- 3° 


- 1993 


7 2 


- 3 


- 309 


- 31435 


a, 2 




+ 2 ' 


+ 37 


e* 


+ 17 


+ 433 


- 343 


eV 2 




- 16 


— 191 


e' 4 









e 2 y 2 


- 3 


— 13 6 


- 13" 


e'V 






+ 35 


r 4 




+ 1 


— 121 


e 2 a, 2 






+ 4 


7 2 ai 2 






— . . . '2 


Sum ... 


+ 1 +127 


+13902 


+ 2369899 



• 



198. Arg. I+2/,]). 



i. 


3 


2 


i 





— 1 


—2 


-3 


-4 


e 


+ 12 


+ 1446 


+ 174865 


+ 22648107 


— 4608089 


-35 221 


— 291 


■ — '2 


• e 3 


+ 10 


+ 574 


+ 20813 


— 8 533 


+ 1231 


- 4i43 


— 114 


— 2 


■ ec' 2 




- 6 


- 238 





+ 1586 


+ 77 


+ 2 




1 ey s 


— 1. 


- 58 


- 3314 





+ 18897 


+ 811 


+ 13 




! ea y 2 






+ 4 


• 9 


- 5° 








e b 


+ 2 


+ 42 


- 44 


+ 5 


+ 4 


+ 4 


- 8 




1 e 3 g /2 




— 2 


— 21 





+ 13 


+ 8 


+ 1 




, e'V 2 




- 17 


- «5 





- 19 


+ 38 


+ 4 




ce' 2 y 2 






+ 6 





+ 15 


- 2 






e/ 






— 2 





- 26 








Sum... 


+ 23 


+ 1979 


+ 191954 


+ 22639579 


-4586438 


-38428 


-393 


-4 















ii2 Mr. Ernest W. Brown, Theory of the 



199. Arg. V + 21J). 



i. 


3 


2 


1 





-1 


— 2 


-3 


e' 


— 1 


— 180 


-21595 


—659271 


— 152090 


-1255 


— 10 


eV 


— 2 


-"3 


- 3476 


- 15490 


- *5"5 


- 636 


-13 


e' 3 






+ 1 


- 5 


+ 11S 


+ 3 




e'y 2 




+ 10 


+ 651 


+ 57°2 


+ 1651 


+ 38 


+ 1 


eV 






7 










eV 


— 1 


— 10 


- 26 


+ 27 


+ 3 


— 44 


- 3 


eV 3 






+ 1 


— 22 


+ 8 


+ 1 




e^e'y'* 




+ 4 


— 2 


+ 120 


+ 82 


+ 14 


+ 1 


e' J y' 2 










— 2 






e'y* 






+ 2 


- 5 


+ 4 






Sum... 


-4 


— 289 


-24451 


—668944 


-165351 


-1879 


-24 





200. Arg. 21J). 




2>\. 


5 


3 


1 


a l 


+ 8 


+ 735 


-125394 


e i a l 


-3 


-383 


- 2433 


e' 2 a, 




+ 7 


— 21 


y z «i 




+ 42 


+ 3°4° 


a[ 3 






— 1 


e 4 a. 


— 1 


- 9 


+ 5 


e 2 y 2 a 1 




+ 10 


+ 36 


7 4 «l 






- 17 


Sum ... 


+4 


+402 


— 124785 











201. Arg. 2 


'+2Z'D. 








i. 


3 


2 


1 





-I 


-2 


-3 


-4 


6* 


+ 2 


+ 169 


+ 13241 


+ 771167 


— 212622 


-31054 


-531 


-7 


e* 


+ 2 


+ 56 


+ 1478 


- 1038 


+ 92 


- 31 


- 59 


— 2 


eV 2 




— 1 


- 25 


- 59 


- 69 


+ 36 


+ 3 




eV 




- 8 


- 299 


- i°37 


+ 95° 


+ 279 


+ 16 




e 2 ai 2 








— 1 


— 2 


— 2 






e e 








+ 1 










eV 




- 3 


- 9 


+ 3 


— 2 




+ 1 




e*y+ 






+ 1 


15 


5 


— 1 






Sum... 


+ 4 


4-213 


+ 14387 


+ 769021 


— 211658 


-3°773 


-57° 


-9 



IfJUl 



mm 

mm 



Ijjj 



■ 






Marion of tin: M,n,n. 



* 13 



202. Arg. l+l' + 2iT>. 



i. 


2 


1 





— 1 


■~2 


-3 


-4 


ee' 


-37 


— 2662 


— 110214 


— 206896 


—4088 


-55 


— 1 


eV 


-16 


- 356 


- 660 


— 149 


- 393 


-19 




eg' 3 






- 84 


+ 87 


+ 6 






ee'y 2 


+ 2 


+ 94 


+ "54 


+ 740 


+ 79 


+ 2 




ee'a x % 




— .1 




— 1 








Sum... 


-51 


-2925 


— 109804 


— 206219 


-4396 


-72 


— I ■ 



203. Arg. l~l' + 2iT>. 



i. 


3 


2 


1 





— 1 


—.2 


-3 


ee' 


+ 3 


+ 216 


+ T3634 


+ T49260 


+ 27878 


+ 578 


+ 8 


e 3 e' 


+ 2 


+ 74 


+ ii77 


+ 61 


+ 3° 2 


+ 74 


+ 3 


ee' 3 






- 13 


+ 104 


- 54 


+ 1 




ee'y 2 




- 7 


- 203 


- *549 


+ 385 


- 16 




ee'a, 2 








+ 2 








Sum... 


+ 5 


+ 283 


+ H595 


+ 147878 


+ 28511 


+ 637 




+ n 1 

1 



204. Arg. 2l' + 2il). 



i. 


2 


1 


0':': 


— 1 


— 2 


-3 


e n 


+ 1 


— 12 


-7313 


—7602 


— 107 


— 1 


eV 2 


~4. 


-182 


— 260 


- 586 


- 48 


— 1 


e'* 






+ 2 


+ 6 










+ 5 


+ 66 


+ 66 


+ 4 




Sum... 


-3 


— 189 


-7505 


-8116 


-151 


— 2 



114 



Mr. Ernest W. Brown, Theory of the 









205. 


Arg. 2F + 2^D. 










i. 


3 


2 


1 







-1 


— 2 


-3 


r 2 




-39 


-4193 


— 409912 


— 


56040 


-53 




eV 


— i 


—42 


— 1614 


- 849 


+ 


754 


+ 68 


+ 1 


c'V 






+ 4 


4- 1 


+ 


57 






y* 




+ I 


+ 54 


- 834 


+ 


48 


+ 9 




yW 










— 


3 






e 4 y 2 




- 6 


+ 4 


4- 11 


+ 


4 






eV 




+ i 


+ 4 


- 3i 


+ 


6 


+ 1 




Sum ... 


— i 


-85 


-5741 


— 411614 




55174 


+ 25 


+ 1 



206. Arg. 1+ 2i x D. 



2!,, 


S 


3 


1 


— 1 


-3 


-5 


ea^ 


+ 1 


4-29 


-8546 


+ 18757 


4-3180 


+ 5 


e 3 a[ 


— 1 


-38 


— 127 


+ 214 


4- 108 


+ 11 


ee'^a. 






+ 6 


+ 8 


- 13 


— 1 


«7 2 «i 




+ 7 


4- 226 


— 425 


- 69 


— 1 


Sum ... 




— 2 


— 8441 


+ I8554 


4-3206 


+ 14 



207. Arg. I' + 2iJ). 



2i,. 


5 


3 


1 


— 1 


-3 


-5 


e'a. 


+ 1 


4-112 


4-17654 


+ 593 


-90 


— 1 


e 2 e'a l 


+ 1 


+ 43 


+ 599 


+ 1 . 


+ 27 




e' 3 a! 






- 3 


+ 3 






e'y 2 a 1 




- 5 


- 258 


- 38 


- 3 




Sum ... 


+ 2 


4-150 


4-17992 


+ 559 


-66 


— 1 



208. Arg. 3I+21D. 



i. 


2 


1 





— 1 


— 2 


-3 


-4 


e? 


+ 17 


+ 983 


+ 36339 


-13273 


— "97 


— 296 


-8 


a' 


+ 5 


4- 106 


- 95 


+ 13 


+ i 


— 2 


— t 


cV 2 




3 


— 8 


~ 5 


— 2 


+ 1 




e 3 y 3 


— 1 


- 26 


— 112 


+ 7 2 


+ n 


+ 4 


| 


Sum . . . 


4-21 


4- 1060 


4-36124 


-13193 


— 1187 


-293 


-9 



Miitiml !•/ I hi: .!/-<. i//. 
209. Arir. 2l+l' + 2i\). 



1 1 



i. 


2 


1 





— 1 


— 2 . ■ 


~~ 3 


-4 


-5 


-268 


-7700 


-8664 


— 2762 


-85 


— 2 


eV 


— 2 


- 33 


- 45 




- 6 


- 8 


— 3 


eV 8 






- 6 




+ 2 






eVy* 




+ 11 


+ 92 


+ 26 


+ 23 


+ 2 




Sum . . . 


-7 


— 290 


-7659 


-S63S 


-2743 


-91 


-3 



210. Arg. 2I—V + 21D. 



i. 


2 


1 





— 1 


-2 


-3 


eV 


+ 26 


+'III2 


+ 9833 


-^-2601 


+ 35 2 


+ 12 


eV 


+ 8 


+ 9 2 


- 13 


+ 3° 


+ 3 


+ 2 


e 2g/3 




— 1 


+ 6 


— 11 


+ 1 




eVy 2 


— 1 


— 21 


— in 


+ 85 


+ 4 




Sum . . . 


+ 33 


+ 1182 


+ 9715 


+ 2497 


+ 360 


+ 14 



211. Arg. I+2V + 21D. 



i. 


1 





— 1 


— 2 


~3 


ee' 2 


— 1 
-13 


— 1 1 70 

- 9 

+ 10 


-7445 
— 10 

+ 24 


- 293 

— 24 

+ 5 


-6 | 
— 2 1 


um . . . 


— 14 


— 1169 


-7431 


— 312 


-8 I 



212. Anr. I— 2V + 21D. 



ilii 



ee' 2 


+ 19 


+ 719 


+ 2615 


+ 2552 


+ 18 


eV 2 


+ 6 


+ 49 


+ 3 


+ 4 


+ 5 


e«' 2 y s 


■ — 1 


- 9 


- 32 


- 17 


— 1 


um .... 


+ 24 


+ 759 


+ 2586 


+ -'5_v- 


+ 22 



llny.M. AijTiMx. Sue.. Vol. LVIL. 



tK 



mt 



1 10 



Y\V. KliM..->T \V. P>Rii\V.\. '/'//. ■■/''/ i if III- 

213. Arg. $l' + 2iD. 



i. 





— 1 


— 2 


1 e' s 
\ eV 3 

g'3 y 2 


- 98 

- 6 

+ 1 


— 326 

— 21 

+ 2 


- 7 j 

■■ :'3 


Sum ... 


-103 


-345 


— 10 1 



214. Arg. I+2F + 21D. 



i. 


2 


1 





— 1 


-2 


-3 


ey* 


— II 


-809 


-45068 


— 242 


-309 


— I | 


aV 


- 7 


— 196 


+ 59 


+ 63 


+ 4 


+ 1 1 


ee' 2 y 2 




+ 1 


+ 1 




+ 2 


! 


ey* 
Sum... 




+ 12 


- 92 




+ 2 




-18 


-992 


—45100 


— 179 


-301 





215. Arg. I— 2F + 2?D. 



1 i. 


3 2 


1 





-1 


-rZ 


-3 


~r -55 


-6331 


+ 39316 


+ 9367 


+ 165 


+ 2 


«3y 2 


-11 


+ 10 


- 239 


+ 24 


+ 40 


+ 1 1 


ee n y 2 




+ 10 


— 22 


- 4 






ey 4 


— 1 


- 7i 


+ 477 


— 21 


- 3 


J 


Sum... 


— 1 —67 


-6382 


+ 39532 


+ 9366 


+ 202 


+ 3 J 







216. Arg. Z' + 2F + 2i"D. 






1 i. 
, e'y 2 


2 


1 


— 1 


— 2 


+ 1 


+ 47 +392 


-2195 


-15 


I eVy 2 


1- 1 


+ 20 + 31 


+ 36 


+ 7 ! 


e'V 






+ 2 


- 


e'y 4 




-1 - 7 


+ 2 


+ l 


Sum ... 


+ 2 


+ 66 +416 


-2155 


— 7 J 



Motion of the Moon. 
217. Arg. I'— 2? + 2/D. 



t. 


'. .1 





— 1 


— 2 




e'y a 


— 1449 


+ 59 


+ 3°4 


+ 6 


■:. : :f 


eVy 2 


+ 4 


+ 24 


+ 83 


+ 5 




e'V 












e'y 4 


+ 3 


- 7 


- 3 






Sum... 


-1442 


+ 76 


+ 384 


+ 11 





1 i: 



218. Arg. 2^+22'jD. 



i 21,. 


3 


1 


— 1 


-3 


-5 


-7 j 




—2 

-3 
+ 1 


-595 
- 6 

+ 17 


+ 1773 
+ 10 

- 38 


+ 1228 

+ 3 
— 10 


+ 57 
+ 3 

— 1 


+ 1 1 


i Sum... 

1 


—4 


-584 


+ 1745 


+ 1221 


+ 59 


I 

+ I 1 



219. Arg. l+l'+2ij). 



2i,. 


3 


1 


— 1 


-3 


1 

-s 


1 «'«'«! 

j <?e'y 2 a, 

!: : 


+ 19 
+ 5 

— 1 


+ 1244 

+ 41 

— 20 


+ 143 

— 2 

— 4 


+ 230 

+ 7 

- 4 




Sum... 

_ , 


+ 23 


+ 1265 


+ 137 


+ 233 


'■':■■ :'■* 



220. Ars. l—F + 2i,~D. 



1 


3 


r 


■ — 1 


-3 . • " 


-s 




ee'aj 
| eVa, 
j ee'y l a x 


+ 6 
-3 


— 122 

- 5 
+ 5 


— 1062 

- Si 
+ 26 


-274 

— 7 
+ 5 


—4 

— 1 
+ 2 




1 Sum... 


+ 3 


— 122 


— 1087 


—276 


-3 





n8 



Mr. Ernest AV. Brown, Theory of the 
221. Arg. 2F + 2« 1 D. 



7 2 «! 


+ i 


+ 254 


+ 584 


+ 258 


+ 1 


eV«l 


+ 3 


+ 6 




- 4 




: y i a l 




- 6 


— 2 


— 1 




Sum... 


+ 4 


+ 254 


4-582 


+ 253 


+ 1 

1 









222. 


Arg. 


4?+2z'D. 








i. 


2 


1 







— 1 


— 2 


~3 


-4 




+ 2 


+ 72 

— 2 


+ *953 

- 5 

— 10 




-957 
+ 5 


+ 1 

+ 2 


— 14 


-4 


Sum... 


+ 2 


+ 70 


+ 1938 




-952 


+ 3 


— 14 


—4 







223. Arg. 2?+2F + 2^D. 






'• 


2 


1 -1 


— 2 


~3 


e 2 y 2 

e 2 y 4 


— 2 

— 1 


-105 -4005 +558 
— 20 + 15 +1 
+ 2 — 6 — 2 


—6 
+ 1 


-3 


Sum... 


-3 


— 123 -399 6 +557 


-5 


-3 



224. Arg. 2I— 2F + 2?T). 



i. 


2 


1 







— 1 




-2 


-3 


— 10 


-45° 


— 


1352 


+ 537 


+ 


171 


+ 4 


eV 


— 1 


— 4 


+ 


80 


- 3 


+ 


3 


+ 1 


e 2 y 4 




- 5 


— 


26 


+ 4 


— 


1 




Sum . . . 


— 11 


-459 


— 


1298 


+ 538 


+ 1 


73 


+ 5 







225. 


Arg. 


4F + 2? 


T>. 




i. 


1 











— I ! 


7 4 


+ 8 
+ 6 






+ 407 

4- 11 

+ 418 




+ 77 

_ 3 | 


Sum... 


+ 14 






+ 74 






: 












r 
"i '~ 

ye' 2 
f 



yV 2 
ye 4 



+ 5 
+ 8 



+ 2 



+ 633 

- J 3 

+ 527 



+ 58 

— 2 



Motion of the Moon. 

226. Al'l'. F+2/'I>. 



+ 94476 

- 876 

+ 24010 

- 92 

+ 3 

— "I 

— 121 
+ t 

— 122 

— 16 



+ 1S5172S3 

O 
-. 558" 

o 
o 

- 9 

o 

o 

+ iS 

o 



— I) 1 344(1 

- 95a 

- 4857 
+ 588 

— 9 

— 2 
+ II 

+ 3 



1 19 



-2 




-2897 


-19 


+ 3 




~ 793 


-16 


+ 11 




+ 7 




- S 


— 2 


+ 2 





Sum... J +15 +1192 — 117262 



+ 18461480 



-623658 -3675 -37 



227. Arg. V + I+21D. 



1 . 

i ** 


3 


2 


1 





— i 


- 2 


-3 


-4 


ye 


+ 1 


+ 140 


+ 13019 


+ 1014212 


-167571 


-6536 


-80 


— 1 


y 3 e 




- 3 


- 137 


- 26 


+ 590 


+ 5 






ye 3 


+ 1 


+ 72 


+ 2285 


— 4001 


+ 357 


- 66 


~I5 




yee' 2 




— 1 


- 19 


■-• ■ ."6' 


+ 49 


+ 16 






yea j 2 






+ 1 




— 1 






■iC-.:'\S:-<\?K'i 


ye' 




+ 6 


- 14 


+ 4 


+ 1 






1 

i 


y3 g 3 




— 1 


- 13 


- 6 


— 2 


+ 1 






y 6 e 








+ 3 








I 


Sum . . . 


+ 2 


+•213 


+ 15122 


+ 1010180 


— 166577 


-65S0 


-95 


-1 ! 



ils 



228. Arg. Y — 1+2I.T). 



i. 


3 


2 


1 





— I 


-2 


-3 


ye 


+ 28 


+ 2600 


+ 201433 


— 997081 


-33m 


—401 


-4 


y 3 e 


— 1 


- 41 


- 997 


- 3755 


+ 213 


+ 4 




ye 5 


+ 14 


+ 45 2 


- 884 


+ 1129 


— 5°4 


- 79 


— 2 


yee n 




- 5 


73 


+ 7 


+ 44 - 


+ 1 




yea 2 






+ 2 




— 1 






ye* 


+ 1 


— 2 


4- 1 


+ 2 


+ 2 


— 1 




y s e s 




- 4 


+ 3 


+ 22 


- 5 


+ 1 




r>e 








- ;■-. 


+ 3 






Sum ... 


+ 42 


+ 3000 


+ 199485 


-999695 


-33359 


-475 


-6 



■ 



120 



ye- 
y 3 e' 



ye" 
yc'a^ 



Sum 



■13 



-24 



Mr. Ernest ^y. Brown, Theory of the 
229. Arg. F + l' + 2iT>. 



— 1002 
+ 19 
- 285 



— 1269 



-6125 
+ 177 
- 55° 
+ 6 



-6492 



- 29443 

- 16 

- 253 
+ 23 



-29689 



-341 
- 77 



- 4 ii 



-3 



—4 



230. Arg. ¥—l f +2iT>. 



i. 


3 


2 


1 





— 1 


—2 


-3 


ye' 


+ 1 


+ 9 1 


+ 6844 


+ 4794 


+ 12073 


+ 89 


+ r 


y 3 e< 




— 2 


- 5 2 


- 158 


+ 18 


— 1 




yeV 


+ 1 


+ 65 


+ 1215 


+ 232 


+ 49 


+ 25 


+ 1 


ye' 3 






- 6 


- 5 








yeV 
















Sum ... 


+ 2 


+ 154 


+ 8001 


+ 4863 


+ 12140 


+ H3 


+ 2 



231. Arg. F 2iJ). 



2!,. 


3 


1 


— 1 


-3 


-5 


y«, 


+ 5 


-5418 


+ 4810 


+ 320 


+ 1 


y 3 «i 


+ 2 


+ 136 


- 89 


+ 6 




ye 2 «i 


-37 


- 74 


+ 73 


+ 26 


+ 2 


ye'V 


+ 1 


— 1 


+ 1 


— 2 




Sum ... 


—29 


-5357 


+ 4795 


+ 35° 


+ 3 



y 3 e 2 
yV 2 



Sum 



232. Arg. 3F4-2z'D. 



-3 



- 92 
+ 1 

- 52 

-143 



-5978 

— 10 

— 311 

— 6299 



— 2277 
+ 2 
+ 87 
+ 3 

-2185 



-65 



+ 2 



-63 





































.)/..//. n ../" 


the Moon. 



233. Ar<j. F + 2I + 21D. 



121 



■ 

i. 


2 


1 





— 1 


— 2 


-3 


-4 


ye* 


+ 20 


+ 1341 


+ 62261 


— 15682 


-638 


-81 


— 2 


y 3 e 2 




- 15 


— 24 


+ 65 


+ 3 


+ 1 




ye 4 


+ 8 


+ 200 


- 3i9 


+ 54 




— I 




ye 2 e' 2 




- 3 


5 


— 2 








Sum... 


+ 28 


+ 1523 


+ 61913 


-15565 


-635 


-81 


— 2 



ye" 

y :i e 2 

ye 4 

yeV 2 



Sum.. 



+ 1 



+ 1 



234. Arg. F — 2^+2 iD. 



+53 
— 1 
+ 8 



+ 2451 

- 25 

- 10 

- 3 



+ 60 



+ 2413 



-1630 

— 2 
+ 6 
+ 2 



— 1624 



-31504 
— 3*4 
+ 46 
+ 9 



-31763 



— 2136 
+ 30 

— • 44 
+ 4 



— 2146 



—42 
+ 1 
- 7 



- 4 b 



„ii*j . 



I 



235. Arg. F + ^ + / / + 22'D. 



i ■ i. 


2 


1 





-1 


— 2 


-3 


J yee' 


-4 


-203 


-5340 


-7502 


-593 


-13 


y 3 ec' 




+ 4 


+ 6r 


+ 22 






i yeV 


'■ — -;2 


- 40 


- 49 


+ 14 


- 7 


■ •.— 2 -. 


1 yee' 3 






- 3 


+ 3 






Sum ... 


-6 


-239 


-5331 


-7463 


— 600 


-i'5' 



236. Arg. F-l~l' + 2iD. 



1 i. 


3 


2 


1 





— i 


—2 


yee' 


+ 5 


+ 303 


+ 8975 


+ 5118 


+ 821 


+ 14 


j y 3 ee' 




- 5 


. — 41 ' 


- 27 


4 




'. yeV 


+ 2 


+ 43 


- 28 


+ 1 


+ 9 


+ 3 


j yee' 3 






- 4 


+ 4 






: Sum ... 


+ 7 


+ 341 


+ 8902 


+ 5096 


+ 826 


+ •7 



ill 



122 



Mr. Ernest W. Brown, Theory of the 
237. Arg. F + l-l' + 2iD. 



i. 


2 


1 





— 1 


-2 


-3 


yee' 


+ 21 


+ 1022 


+ 6848 


+ 769 


+ 170 


+ 3 


y 3 ee' 




- 9 


— 80 


+ 17 






yeV 


+ 9 


+ 129 


- 16 


+ 11 


+ 1 




yee' 3 




— 1 


+. 4 


— 2 






Sum . . . 


+ 3° 


+ 1141 


+ 675 6 


+ 795 


+ 171 


+ 3 



238. Arg. Y-I+F + 21D. 



i. 


3 


2 


1 





— 1 


— 2 


-3 


yee 


— 1 


—44 


— 1302 


-5707 


— 1762 


-50 


— 1 


y 3 ee' 




+ 1 


— 12 


+ 42 


+ i7 


+ 1 




yeV 




- 8 


— 11 


+ 15 


- 29 


— 9 




yee' 3 






+ 2 


- 5 


+ 1 






Sum . . . 


— 1 


-5i 


-i3 2 3 


-5655 


-1773 


-58 


— 1 



239. Arg. ¥ + 2l' + 2iD. 



z, 


1 





— 1 


-2 


ye' 2 


— 1 


-49 


-1085 


-25 


y'V 2 




+ 3 


— 1 




yeV 2 


-15 


— 10 


— 10 


— 4 


Sum 


-16 


-56 


— 1096 


— 29 



240. Arg. F-2l' + 2iD. 



i. 


2 


1 





— 1 


ye" 


+ 8 


+ 343 


+ 16 


+ 126 


yV 2 




- 3 


— 1 


+ 1 


yeV 2 


+ 5 


+ 47 


+ 4 


+ 9 


Sum 


+ x 3 


+ 387 


+ 19 


+ 136 



+ 1 



+ 1 



Motion of the Moon. 
241. Arg. F + I+2IJ). 



1 ^ 



m 



yea i 

7« 3 «i 1 


'-3 

+ 1 

••■•-s 


-678 
+ 19 

*""* 7 


+ 439 
- 18 
+ 8 


+ 3°6 
— 2 

+ 2 


+ 10 

+ 1 
+ 1 


Sum ... | 


-7 


-666 


+ 429 


+ 306 


+ 12 



242. Arg. F-I+21J). 



21,. 


S 


3 


I 


-1 


-3 


yen, 
y s eu, 
ye 3 o , 


— 1 

— 1 


— 204 

+ 4 

- 8 


+ 136 

+ 6 

O 


+ 587 
+ 1 
+ 3 


+ 33 
+ 2 


Sum 


' ' ' ■ -*2 


~-2o8 


+ r 39 


+ 59i 


+ 35 



■ 



243. Arg. F + P+21J). 



ye'a 
yVa, 
yeV«, 



I Sum... 



3 


1 


- 1 


'~% 


+ 9 

+ 5 


+ 795 
— 12 
+ 21 


+ 14 

— 1 


+ 23 
+ x 
+ 2 


+ 14 


+ 804 


+ 13 


4-26 



ye'a, 
yVri, 
ye'-'c-'a. 



Sum. 



244. Arff. F-/ / + 2? ] D. 



+ 2 



— 20 
+ 2 

-18 



Royal Astron, Soc, Vol. LVII. 



-788 
+ 10 
- 28 

—806 



-32 



-34 



■ 
Mm 



124 



Mr. Ernest W. Brown, Theory of the 
245. Arg. 3F + Z+2^D. 



i. 


1 





— 1 


-2 


-3 


y 3 e 
y 5 e 
•v*e 8 


-23 
- 8 


- 992 

- 2 

- 27 


-343 
+ 1 
+ 13 


+ 6 

+ 1 


— 1 


Sum 


-3i 


— 1021 


-329 


+ 7 


— 1 



246. Arg. 3F-I+2H). 



j. 


2 


1 





-1 


— 2 


y 3 e 
y 6 e 
y 3 e 3 


-s 

— 2- 


-234 
+ 1 
— 11 


-2808 

- 39 
+ 33 


+ 290 

+ 3 
— 1 


+ 5 


Sum 


-7 


— 244 


— 2814 


+ 292 


+ 5 



247. Arg. F + 3I+21D. 



i. 


2 


1 





-1 


-2 


3 


- 4 


ye 3 

i 7 3e3 

j ye 5 


+ 2 
+ 1 


+ 124 
— 2 
+ 17 


+ 4015 

- 4 

- 27 


-1528 
+ 6 
+ 6 


+ 8 
+ 1 


-7 


— T 


i Sum 

1 


+ 3 


+ 139 


+ 39 8 4 


-1516 


+ 9 


-7 


— I 



248. Arg.F-3Z+2*D. 



i 


4 


3 


2 


1 





-I 


-2 


ye 3 
y 3 e 3 
ye 5 


+ 1 


+ 32 
— 1 


+ 22 
— 1 


+ 2 S3 
+ 2 


-1570 
- 19 
+ 4 


— 146 

+ 3 

- 4 


-4 


Sum ... 


+ 1 


+ 31 


+ 21 


+ 255 


-1585, 


-147 


-4 



Motion of the Moon. 
249. Arg. 21D. 



<^5 



j 


3 


2 


1 


1 


I 


+ 11 


+ 1568 


+ 245748 


+ 34226987 


e 2 


+ 16 


+ 98S 


+ 37988 





e' 2 




- S 


— 232 


+ 11 


¥■ 




— 12 


- 9 2 4 





a * 2 






+ 4 




e 4 


+ 5 


+ 86 


- 40 


+ 1 


eV 2 




- 5 


25 





<?V 




— 10 


- 187 


+ 3 


e'y 











7 1 






+ 1 


— 1 


Sum . . 


+ 32 


+ 2607 


+ 282333 


+ 34227001 



250. Arg. I+2H). 



i. 


3 


2 


1 





-1 


— 2 


3 


-4 I 


e 


+ 3 


+ 3°5 


+ 27534 


+ 1866057 


+ 345043 


+ 5396 


+ 61 


+1 ! 


e 3 


+ 3 


+ 126 


+ 3493 


681 


- 128 


+ 678 


+ 25 


+1 


ee' 2 




• ■ - : '— ; - 2 


- 39 


- 13 


- 118 


— 11 






:..ey*. ;■ 




3 


— 104 


+ 3 2 


- 1687 


- 47 


— 2 




ea^ 






+ 1 




+ 4 








e s 


+ 1 


+ 10 


- 6 






— 1 


+ 2 


1 


«V 




- 3 


- 18 


— 10 




— 7 






Sum... 








+ 13 


+ 3 








+ 7 


+ 433 


+ 30861 


+ 1865398 


+ 343 rl 7 


+ 6008 


+ 86 


+2 j 



251. Arg. /' + 2/D. 



*• 

1 e ' 


2 


t 





'";■■— t. 


2 


— 3 


-33 


-2569 


— 3924 


+ I74I5 


+ 223 


+ 2 1 


I e*e> 


— 20 


- 45i 


— 127 


+ 1874 


+ 120 


+ 4 


e' 3 






+ 2 


— 14 


— 2 




i e y 




+ 17 


+ 47 


- 73 


— 2 




eV 




— i 








j 


Sum ... 


-53 


—3004 


— 4002 


+ 19202 


+ 339 


+ 6 



126 



Mr. Ernest W. Bkown, Theory of the 
252. Arg. 2? 1 D. 



e' 2 a l 
y 2 a, 



Sum 



+ 1 



+ 1 



+ 81 
-62 
+ 1 
+ 3 



+ 23 



1 

— 9800 

— 192 

— 2 
+ 242 



-975 2 











253. Arg. 2 


1 + 


21D. 








?'. 


3 


2 


1 







- 1 


-2 


-3 


-4 


e 2 




+ 4i 


+ 2546 


+ 101788 




-3052 


+ 3760 


+ 98 


+ 1 


e 4 


+ 1 


+ 13 


+ 299 


- 125 




+ I 


+ 4 


+ 11 


+ 1 


eV 2 






— j 


- 9 






- 5 






e 2 y 2 






- 9 


+ 3 




+ 12 


- 37 






Sum... 


+ 1 


+ 54 


+ 2833 


+ 101657 




-3039 


+ 3722 


+ 109 


+ 2 



ee'y 2 



Sum... 



_ 4 



— 426 

- 61 

+ 2 
-485 



254. Arg. l + l' + 2iD. 



-9536 

- 56 

- 7 
+ 97 

-9502 



+ i45 I 5 
+ 7 

- 5 

- 62 



— 2 


"~ 3 


+ 619 


+ 11 


+ 63 


+ 4 | 


— 2 




- 6 


1 



+ 14455 



+674 



+15 



255. Arg. I-V + 21D. 



i. 


3 


2 


1 





- 1 


- 2 


•-3 


ee' 


+ 1 


+ 45 


•f 2122 


+ 11654 


— 2214 


- 90 


-3 


■■ eh' 




+ 15 


+ 195 


+ S 


— 24 


— 12 


— 2 


ee' 3 






- 4 


4- 7 


+ 4 






ee'y 2 


+ 1 




- 8 


— 124 


- 26 






Sum... 


+ 60 


+ 2305 


+ 1 1542 


— 2260 


— 102 


-5 



'.. y- 



■Sum ... 



256. Al'g. 2/' + 2/'D. 





I 





— 1 


-2 




— 2 
-26 


-S 4 

- 4 
+ 2 


+ 853 
+ 7i 
- 4 


+ 19 
+ 9 




-28 


-86 


+ 920 


+ 28 



12; 



257. Arg. 2F + 2&T). 



i. 


1 





-1 


-2 


-3 
+ 1 


7 2 


+ 9 


+ 667 


— 1 07 1 


+ 41 


eV 


-17 


-788 


+ 16 


- 9 




e'V 






+ 2 






V 4 


— 1. 




+ 1 


— 1 




um ... 


- 9 


— 124 


— 1052 


+ 3i 


+ 1 



258. Arg. I+21J). 



2!",. 


3 




1 


-.1 


-3 


eaj 


+ 2 


— 


1 103 


+ 120 


-378 


e 3 a t 


-7 


— 


16 


+ 2 


- 15 


e-Ai 


+ 2 


+ 


29 


— 4 


+ 8 


Sum... 


-3 


— 


1090 


+ T18 


-3*5 



-3 



■259. Arg. I' + iiJ). 



2J. 


3 


1 


— i 


~~ .' 


e'a, 

1 


+ 20 


+ 1464 


-40 


+ 10 


e 2 e'a. 


+ 7 


+ 5° 




- 3 1 


«V«i 




22 


+ 3 




>Sum ... 


+ 27 


+ 1492 


-37 


+ 7 | 



128 



Mr. Ernest W. Bbown, Theory of the 



260. Arg. 3I+21T). 



i. 


2 


1 





— 1 


-2 


-3 


-4 


e* 


+ 5 


+ 219 


+ 6231 


— 1192 


+ 76 


+ 47 


+ 2 


e-' 


+ 2 


+ 24 


- is 


+ 1 




+ 1 




e 3 y 2 






— 1 


+ 4 


— 2 


— 2 




Sum ... 


+ 7 


+ 243 


+ 6215 


-1187 


+ 74 


+ 46 


+ 2 



261. Arg. Z+2F + 2z'D. 



! 

i. 


1 





-1 


-2 


-3 




ey* 


+ 1 


+ 63 


-847 


+ 15 


+ 2 


e 3 y 2 




-73 


+ 11 


— 1 






ey 4 






+ 3 






. 


Sum... 


+ 1 


— 10 


-833 


+ 14 


+ 2 





262. Arff. /— 2F + 2/D. 



i. 


2 


1 





— 1 


— 2 

— 1 


ey 2 


— 2 


-476 


— 7063 


- 88 


<3 3 y s 


-3 


— 1 


+ 29 


— 24 




ey 4 




_ 4 


— 102 




— 1 


Sum ... 


-s 


-481 


-7136 


— 112 



m 



Mill,,,, „f //,(■ .]/, ,.,!,. 



120 



Section (v). The Final Values of the Coefficients in Longitude, Latitude, and Parallax. 

263. The following tables, giving the final values of the coefficients, are arranged, 
first, according to the order and composition of the principal characteristics ; second 
according to the signs of the multiples of I, I', F : and third, according to multiples of 
1) in descending order. 

In the first column, headed " P. C," is given the principal characteristic. In the 
second, third, fourth, and fifth columns, headed " /." " /'," "J>\" « ]) » are O .[ von t ] ie 
multiples of those arguments (Dklauxay's notation) to which the coefficients in the last 
column correspond. The characteristic is understood to belong to all coefficients down 
to the next printed characteristic ; a similar remark applies to the multiples of /, If, F. 

As stated earlier, the system of axes used is the same as that in which Delauxay 
expressed his final results. 

These coefficients are all definitive for the corresponding arguments, with the 
following exceptions : — 

(a) Small changes due to possible changes in the values of the arbitrary constants. 

(/3) Small additions due to the terms arising from the perturbations noted in (a), 
{!>), ((■) of § 4, Chap. I. These are very minute and are easily obtained. They are, 
however, more simply treated by the method of the variation of arbitrary constants^ 
and will therefore be given with the treatment of the planetary inequalities. 

The results have been discussed, and a comparison has been made with those of 
Hansen elsewhere.* 



* E. W. Brown, " The Final Values of the Coefficients in the New Lunar Theory," Monthly Notices, 

January 1905. 



HO 



Mr. Ernest W. Brown, Theory of the 



264. Longitude. Coefficients of Sines. 



P.O. J 


I. 1'. F. D. ; 


Coeff. 


I 


OOO 8 : 


+ O'OOI 




6 


+ -127 




i 

4 


+ i3"9° 2 


1 


2 


+ 2369-899 


e 


1 6 


+ -023 




4 


+ 1-979 




2 


+ i9 r 954 







+ 22639-580 




— 2 


- 4586-438 




-4 


- 38428 




-6 


- - 393 




-8 


•004 


e' 


010 6 


— '004 




4 


- -289 




2 


- 24-451 







— 668-944 




— 2 


- 165-351 




—4 


1-879 




-6 


— '024 


"i 


0005 


+ -004 




3 


+ -402 




1 


- 124-785 


e- 


200 6 


+ '004 




4 


+ -213 




2 


+ 14-387 







+ 769-021 




— 2 


— 211-658 




—4 


- 3°-773 




-6 


'570 




-8 


— -009 


ee' 


1 1 4 


- -051 




2 


| - 2-925 







j — 109-804 




! -2 


— 206-219 




—4 


- 4-396 




-6 


— -072 




-8 


1 — -ooi 



P.O. | 

i 


/. V. F. D. 


Coeff. 


ee' | 


1 —1 6 


+ 0-005 


I 


4 


+ -283 


{ 


2 


+ 14-595 







+ I47-878 




— 2 


+ 28-511 




-4 


+ -637 




-6 


+ -on 


e' 2 


0204 


— -003 




2 


- -189 







- 7-505 




— 2 


- 8-n6 




-4 


- -151 




-6 


— "002 


7 2 


0026 


— 'OOI 




4 


- -085 




2 


- 5-741 







—411-614 




— 2 


- 55-174 




-4 


+ -025 




-6 


+ -ooi 


e'cq 


1003 


— '002 




1 


- 8-441 




— 1 


+ 18-554 




-3 


+ 3-206 




-5 


+ -014 


e'aj 


0105 


+ -002 


1 


3 


+ 'IS 




1 


+ I7'992 




— 1 


+ -559 




-3 


- -066 




-5 


— "OOI 



.1/.. '!••!,- ■■/' '''". M"->ii. 



131 



Longitude. Coefficients of Sines (continued). 



: P.O. 


j I. 1'. P. D. 


r.vff. 


. e 3 


1 

; 3 4 


+ 0'02I 


■ 


2 


+ ro6o 


\- 





+ 36-124 


i 


— 2 


-I3T93 


: fe': : ';v>:0>.v : o> 


—4 


— 1-187 


:'l?Sxo:' ;: ' :: ^v:^ 


-6 


- '293 




-8 


— '009 


: eV 


210 4 


— -007 


' 


2 


— '290 


1 





- 7'6S9 




— 2 


- 8-638 


§>ft?SSS-:X? 


—4 


- 2-743 




-6 


— -091 




-8 


- -003 




2 —1 4 


+ "033 




2 


+ 1-182 







+ 9*7*5 


i 


— 2 


- 2-497 


| 


-4 


+ '360 


i 


-6 


+ "014 


! ee' 2 


1202 


— '014 


i 





— 1-169 


; 


— 2 


- 7'43i 




-4 


- "3 "2 1 


j 


-6 


— -008 i 




1 —2 4 


+ -024 




2 


+ '759 


! 





+ 2-586 


< 


— 2 


+ 2"539 




-4 


+ '022 


c '3 


0300 


- t°3 




— 2 


- -345 | 


1 


-4 


— "010 



P. c. 



"'. P. D. 



Coeff. 



■ -, ' 



'■V 



e 2 a! 



024 

2 

o 

— 2 

-4 

—2 6 

4 
2 
o 

— 2 

-4 
-6 

1 24 

2 
o 

— 2 

-4 

1—2 2 

o 

— 2 

— 4 
003 

1 

— 1 
-3 
-5 
-7 

1 o 3 
1 

— 1 
-3 
-5 



— C018 

— "992 
— .\y\:z 

— -i79 

— -301 

— -ooi 

— -067 

- 6-382 

+39-532 
+ 9-366 . 

+ '202 



+ 
+ 
+ 



•O03 

•0G6 



+ '416 

- 2-iS5 

- '007 

- 1-442 



+ 
+ 

+ 



•076 

■3M 

•on 

— -004 

- -584 
+ 1745 

+ I-22I 

+ '°59 

+ 'OOI 

+ -023 

+ 1*265 

+ -137 
+ -233 

+ 'OOI 



llll 



mm 

mmm 
111111 

8 



Royal Astron. Soc, Vol. LV1I. 



132 



Mr. Ernest W, Brown, Theory of the 



Longitude. Coefficients of Sines (continued). 



P.O. 


I. 1'. 


F. D. 1 


Coeff. 


ee'a x 


i —i 


o 3 


+ 0'003 






i 


— '122 






— i 


-1-087 






-3 


— -276 






-5 


- "003 


e' 2 n. 


O 2 


° 3 


— '002 






i 


- '°39 






— i 


— -042 






-3 


— - oo6 


7 2 «i 


o o 


2 3 


+ '004 






i 


+ '254 






— i 


+ -582 






-3 


+ '253 






-5 


+ - ooi ; 


e* 


4 o 


o 4 


+ "002 






2 


+ '070 






O 


+ 1-938 






— 2 


- - 95 2 






— 4 


+ -003 






-6 


— "014 






-8 


— "004 


I eV 


3 i 


O 2 


— '° 2 S 






O 


- - 552 


1 




— 2 


- -483 


1 




— 4 


— TOO 


j 




-6 


- - °39 






-8 


— "OOI 




3 -i 


o 4 


+ '°°3 






2 


+ -088 









+ -682 






— 2 


- -183 






-4 


— "029 






-6 


+ '005 



P.O. 



e'e ■ 



F. D. 



eV 



1 -3 



o o 

— 2 

-4 

-6 

o 4 

2 

o 

— 2 

—4 

-6 

o o 

— 2 

-4 

o 4 
2 
o 

— 2 

o o 

— 2 
2 4 

2 
O 

— 2 

-4 
-6 

-2 4 
2 


— 2 

-4 
-6 



Coeff. 


— C067 


— 


•298 


- 


•161 


— 


•008 


+ 


•003 


+ 


•062 


+ 


•197 


+ 


•255 


+ 


■036 


+ 


•001 


— 


•018 


— 


•250 


- 


•016 


+ 


•001 


+ 


•032 


+ 


•051 


+ 


•003 


— 


■001 


— 


•013 


— 


•003 


— 


•123 ! 


-3 - 996 


+ 


•557 ! 


- 


•005 j 


— 


•003 


— 


•on 


— 


'459 


— ] 


[•298 


+ 


•538 


+ 


•173 


+ 


•005 



T 



Million hi ihr Mr.,,, i. 



'33 






Longitude. Coefficients of Sines {continued). 






: 



P.C. 


J. J'. F. D. 


Coeff. 


ee'y 2 


I I 2 2 


+ 0'0I2 




o 


+ "263 




— 2 


+ -059 




. -4 


— '024 




-6 


— 'ooi 




I I —2 4 


4" '002 




2 


+ -083 




o 


- -083 




— 2 


+ '4-'7 




-4 


+ -019 




I —I 2 4 


— -002 




2 


— '064 







- '3°4 




-2 


4- '002 




-4 


4- -018 




I —I —2 4 


— '00 7 




2 


- - 372 




■ o 


+ -083 




— 2 


- -065 


j 


-4 


— - 002 


| e'V 


O 2 2 O 


+ '004 




— 2 


— "066 


" •• 


-4 


— - 002 




O 2 —2 2 


— '025 







— - 002 




— 2 


4- -016 


V 4 


4 2 


4- '014 







4- -418 




— 2 


+ "074 


e 3 a t 


3 1 


— "042 




— I 


+ "130 




-3 


+ -045 




-s 


4- - oi6 




-7 


+ "OOI 



P C. 


I. 


I'. 


F. D. 


Coeff. 


eVa ! 


2 


1 


P 3 
1 

— 1 
~3 

-5 


4-o - oo3 
4- -092 
4- - oo6 
+ "084 
4- '006 




2 


— 1 


1 
— 1 
-3 
-5 


— "014 

— '3S 2 
+ "042 

— -003 


ee n a x 


I 


2 


1 
— 1 
-3 


— -008 

— 'O02 

-»• "012 




I 


— 2 


-1 

-3 


+ - o°3 
4- "ooi 


e' 3 a l 


O 


3 


■ I 
— x 


— 'OOI 

— '002 


ey 2 oi 


I 





2 1 
-— 1 
-3 

-5 


+ -045 
+ -024 
+ -030 

4- "002 




1 





-2 3 

1 

— 1 

-3 


— - OIO 

— '041 

— '016 

— "OTI 


e'y^a s 


; 


• ' i ,' 


2 3 
1 

— 1 
-3 


— "OOI 

- "035 

+ - oi3 

+ '020 




:. o : 


1 


-2 • 3 

1 

— 1 


4- '009 

— 'OOI 

— - 002 



m 



134 



Mr. Ernest W. Brown, Theory of the 



Longitude. Coefficients of Sines (continued). 



P.O. 


i. 


I'. 


F. D. 


Coeff. 


e'° 


5 


o 


O 2 
O 

— 2 

-4 


+ o'oo5 

+ 'US 
— '069 
+ '004 


eV 

i 


4 


i 


O 2 
O 

— 2 

-4 
-6 
-8 


— '002 

— -040 

— -030 

'000 

— "002 

— 'OOI 


! 


4 


— i 


O 2 

o 

— 2 

— 4 


+ -007 

+ -048 

— '019 

— - 00I 


•J /o 

e 6 e " 


3 


2 


o o 

— 2 

— 4 
-6 


— -003 

— "016 

— -006 

— '003 




3 


— 2 


O 2 
O 

— 2 

-4 
-6 


+ '°°5 
+ '016 

+ - OII 

+ '004 
4- '001 


eV 3 


2 


3 


o o 

— 2 

—4 
-6 


— 'oo r 

— '010 

— -008 

— 'OOI 




2 


-3 


O 2 


— 2 


+ '°°3 
+ '004 

+ 'OOI 


"\ 9 

& y 


3 


o 


2 4 
2 
O 

— 2 


— -003 

— 'Oil 

- - 33° 

+ -092 



P. C. 


I. 


I'. F. D. 


Coeff. 


3 


—2 4 


— cool ! 






2 


- . '°33 | 









- -055 






— 2 


— '005 






—4 


+ -009 






-6 


+ "003 


e'Vy 2 


2 


122 


+ '002 









+ - °43 






— 2 


+ -028 




2 


1 —2 2 


+ '009 









4- "026 






— 2 


4" '022 






-4 


+ '016 






-6 


+ 'OOI 




2 — 


122 


— '009 









- '053 






— 2 


+ "004 




2 — 


1—2 4 


— 'OOI 






2 


— '029 









— '024 






— 2 


•000 






-4 


— '002 


ee'V 


1 


220 


+ "003 






— 2 


+ '004 






-4 


— 'OOI 




1 


2 —2 2 


— '002 









'OOO 






— 2 


+ -015 






-4 


+ 'OOI 




1 — 


222 


— '003 









— -005 






— 2 


+ -007 






-4 


— 'OOI 



\rnr-. 



IB 



M>Al"H <■/ !/», M<vi< 



135 



Longitude. Coefficients of Sines (concluded). 



I- 



i 
I' 



'••'•• 


■/. 


.•• .vl 


F. D. 


Coeff. 




r. c. 


I. 


V. F. D. 


Coeff. 


t .,,'i y l 


1 


— 2 


— 2 4 


— O'OOI 


e i y-a l 


&M-- 


0-2 3 


— - OOI 








2 


— - oi6 








1 


— 'OOI 











•000 








— I 


+ 'OOI 








?— 2 


— '005 








-3 


— '003 


e '3 y 2 





■3 


2 —2 


— - 002 




ee'y' 2 a x 


I 


1 ■ 2 1 


— "006 







3 


— 2 —2 


+ 'OOI 








— 1 


+ 'OOI 


ey 4 


1 





4 2 


+ '003 








-3 


+ "002 











+ -090 






I 


1 —2 1 


— "002 








— 2 


+ '009 








— 1 


*000 




1 





-4 4 


— "OOI 








-3 


— - OOI 








2 


+ "OOI 






I 


-1 2 -3 


— - 00I 











- -080 






I 


-1 —2 3 


— 'OOI 








— 2 


— '019 








1 


— -004 


e'v 4 





1 


4 

— 2 


— 'OOI 

+ *oo3 








— 1 

-3 


•000 

+ 'OOI 







1 


— 4 2 



+ '002 

•000 




y*«i 





04 1 
— 1 


— "OOI 

— "OOI 








— 2 


— "OOI 




e° 


6 


000 


+ -007 


e 4 a! 

> 


4 





1 
— 1 
-3 

-s 


— -003 

+ *OIO 

+ "002 
+ "OOI 




«v 


4 


022 


— 2 
-4 


— "OOI 

— -025 

+ 'OIO 

— -ooi 


| 

i 
1 


3 


1 


1 
— 1 
-3.' 


+ -007 

— 'OOI 

+ -003 






4 


—2 2 


— 2 


— "OOI 

— -007 

+ - 002 








-5 


-f- - 002 




ey 


2 


042 


+ "OOI 




3 


— 1 


1 


— *002 











+ -on 


] 






— 1 


- -023 






2 


—4 


+ "OOI 


| 






-3 


+ "007 








— 2 


- -003 


I e 2 y 2 ai 


2 





2 1 
— 1 


+ - oo6 

— -003 








-4 


— '001 


! 













111 

(11 



Ijjjjjgl 



111 



mm 



jjijjj 



mBSm 

mm 

t|jj 

ilBl 



1§ 
m 

Wi 

m 



m 



m 
I 



illP 

JUl 



136 



Mr. Ernest W. Brown, Theory of the 



265. Latitude. Coefficients of Sines. 



P.O. 


I. V. F. D. 


Coeff. 


r 


0016 


+ 0-015 




4 


+ 1-192 




2 


+ 117-262 







+ 18461-480 




— 2 


- 623-658 




-4 


3-675 




-6 


- -037 


ye 


1 1 6 


+ -002 




4 


+ -213 | 




2 


4- 15-122 







+ 1010-180 




-2 


- 166-577 




-4 


6-580 




-6 


•095 




-8 


— -ooi 




— i 1 6 


+ -042 




4 


+ 3-000 




2 


+ 199-485 







- 999-695 




— 2 


- 33-359 




—4 


- '475 




-6 


— -006 


ye' 


1 1 4 


— -024 




2 


— 1-269 







— 6-492 




— 2 


— 29-689 




—4 


— -418 




-6 


— -006 




—1 16 


+ *002 




4 


+ -154 




2 


+ 8-001 







+ 4-863 




— 2 


+ 12-140 




-4 


+ -H3 




-6 


4- '002 



P.O. 


1. 


v. 


F. D. 


Coeff. 


y«i 








1 3 

1 

— 1 

-3 
-5 


— 0-029 
~ 5-357 
+ 4-795 
+ -35° 
+ "003 


y 3 








3 4 

2 



— 2 


— -003 

- -143 

— 6-299 

- 2-185 



ye' 



yee 



-4 
-6 



2014 
2 

o 
— 2 



—4 

-6 

-8 

-2018 

6 

4 
2 
o 

— 2 

—4 

-6 

1 1 1 4 

2 

o 

— 2 

-4 
-6 



— -063 

— -ooi 
+ -028 

+ 1-523 
+ 61-913 

-I5-565 

— -635 

— -081 

— -002 
+ "OOI 

+ -060 

+ 2-413 

— 1-624 

-31763 

— 2-146 

— -048 

— -ooi 

— -006 

- -239 

- 5-331 

- 7-463 

— -6oo 

— -oi5 



^m. 



M 
111 



Mi'ft'on of tin Moan. 



i.S7 



Latitude. Coefficients of Sines (continued). 



m 



■p 



1 



m 



m 
I 



I 



p. ('. 


I. 1'. F. D. 


Coeff. 


j y ee< 


■ — i — i i 6 


+0-007 


j 


4 


+ "34i 




2 


+ 8 - 9<D2 




O 


+5-096 


| 


— 2 


+ -826 


J 


-4 


+ -017 




i —i i 4 


+ '030 




2 


+ 1-141 




O 


+ 6756 


W''-i'-% 


• —2 


+ 79S 


%::■ :%B.. 


—4 


+ -171 


f ; ^S:'¥. 


-6 


+ '003 


: H-' .!'•;:;';-.'" :"; 


— i i i 6 


— 'OOI 




4 


- -051 


u:;^' : W:- ■ 


2 


-i-323 


1 


O 


-S-6SS 




—'2 


-1-773 




-4 


- "OSS 




-6 


— -ooi 


ye' 2 


2 1 2 


— - oi 6 


1 





- -°5 6 




— 2 


— 1 '096 


i 


-4 


— '029 


. 


O —2 I 4 


+ -013 


K : :': : ; : : /^: 


2 


+ -387 







+ -019 




— 2 


+ -136 




-4 


+ 'OOI 


yea. 


I o I 3 


— -007 




1 


- -666 




— i 


+ -429 




-3 


+ -306 




-5 ! 


+ -012 

1 



P. C. 


I. 1'. F. D. 


foeff. 


yea, 


-1 1 5 


— - 002 




■■■■■:. 3 


— -208 




I 


+ -I 39 




1 


+ '59' 




-3 


+ '035 


ye'a, 


11 3 


+ -014 




1 


+ "804 




— 1 


+ -013 




-3 


+■■ '026 




—1 1 1 


- -018 




— 1 


- -806 




-3 


- -°34 


y 3 e 


1 032 


- "031 







— 1 "02 1 


! 


— 2 


- -329 




-4 


+ '007 




-6 


— -ooi 




-1034 


— -007 




2 


l;: i:; .'; ? ?li<|: 







— 2-814 




— 2. 


+ -292 




-4 


+ -005 


yV 


01 3 a 


+ -on 




— 2 


- -°93 




—4 


— -006 




—1 3 2 1 


— -007 







+ -ooi 




— 2 


+ -056 


1 


-4 | 


+ -003 



■ 



jjj 

tlBl 



i38 



Mr. Ernest W. Brown, Theory of the 



Latitude. Coefiicients of Sines {continued). 



P. C. | 


1. V. F. D. 


Coeff. 




P. C. 


I. 1'. 


F. D. 


Coeff. 


ye 3 


3014 


+ 0-003 


yeV 


— 2 1 


1 6 


— O'OOI 




2 


+ '139 








4 


— '029 







+ 3'9 8 4 








2 


+ '056 




— 2 


-1-516 











- '303 




-4 


+ -009 








— 2 


— -129 




-6 


— -007 








-4 


— -005 




-8 


— 'OOI 




yee' 2 


1 2 


1 


- '°55 




—3 01 8 


+ -ooi 








— 2 


— -272 




6 


+ '031 








—4 


- '°34 




4 


+ "021 








-6 


— -ooi 




2 


+ '255 






— 1 —2 


1 6 


+ -ooi 







-1-585 








4 


-f -02 2 




— 2 


- -i47 








2 


+ '319 




—4 


— -004 











+ -062 


yeV 


2 1 1 2 


— -027 








— 2 


+ - oo6 







— -644 






1 —2 


1 4 


+ *002 




— 2 


- -657 








2 


+ -054 




-4 


- '053 











+ -117 




-6 


— -on 








— 2 


+ -107 




— 2—1 1 6 


+ -009 








-4 


+ -004 




4 


+ - 2I7 






— 1 2 


1 4 


— -ooi 


1 


2 


- -063 








2 


- -115 


! 





+ -314 











— -096 




_ *> 


+ -063 








— 2 


— -069 




-4 


+ '001 








-4 


— '003 




2 -- 1 1 4 


+ '003 




ye' 3 


3 


1 —2 


- '037 




2 


+ -114 








-4 


— - 00I 







+ "809 






-3 


1 2 


+ -014 




H_ O 


- -084 




7 3 «i 





3 1 


+ -006 




-4 


+ '002 








— 1 


+ '032 




-6 


+ '002 








-3 


+ -oio 



\f, ,//',, n >>/' tin MdiO) 



\Vj 



Latitude. Coefficients of Sines (continued). 



P. c. 


I. 


V. 


F. D. 


Coeff. j 




2 


6 


i 3 

— i 
-3 

: -.5 : . 


— o-ooi 1 
- -065 

+ "112 

+ '039 
+ "oos 


p:"'' : - : 
1 


— 2 


Q 


i S 
3 

, . i 
— i 
-3 


- -oos | 

- -049 

- -078 
+ -036 
+ '003 , 


y«e'a , 


I 


I 


i 3 

i 
— i 

—3 

-s 


+ '002 
+ •101 
— -on 

+ '021 
+ 'OOI 




— I 


— I 


i 3 
i 

.' — * 
-3 


•/•- "OI3 ■:' 
+ 'OOI 

- '034 

— "004 


i 
| 


I 


—'I 


i i 
— i 
~3 

-s 


- -009 

— *oo6 1 

- -013 

— 'OOI 


1 

V- 


• ' — I 


I 


' 3 

I 

— I 

-3 


+ '022 

- "°S6 

+ "020 1 
+ '002 [ 


yc n , H 


O 


2 


i i 

— i 

-3 


— \002 I 

— '002 
+ 'OOI 




O 


— 2 


i i 


■ + '002 1 








— i 

i 


+ '002 ! 



P.O. 


1. 


V. 


F. D. 


Coeff. 


r' 








S 


+ o'oo5 

+ - 002 


•y 3 e 2 


2 





3 2 



— 2 
.—4 • 


— '004 

— 'ii6 

— '022 

+ "005 


I 

I- : ■.■■■'.' 
1 

| 


' . — 2 


' ■ : 


3 4 

' .2 



— 2 

;-4 


— - oo6 

— - o66 
+ -130 

+ 'OTO 
+ 'OOI 


i -fee' 


1 


•■ I. 


3 

—2 


+ -007 

— 'Oil 




— 1 


— $.-; 


3 2 


• ~2 


— -on 
+ '008 




1 


— t 


3 2 


__2 


— '002 

— 'O06 
+ 'OO3 


[■ : 


— 1 


I 


3 2 



— 2 


+ *002 
— '009 
+ '017 


i 7V 2 


O ; 


2 


3 - 2 
-4 


- '003 

— 'OOI 




O 


— 2 


3 - 2 


+ 'OOI 


ye* 


4 





I 2 


. 2 

-4 ; 


+ 'Oil 

+ -266 

- -135 
+ '007 




-4 


O 


1 6 

.■ ..... •... ... ■... ,-.■..; 

4 i 

2 




+ 'OOI 

•000 

+ -025 ; 
— -091 








. —2 • 


— '010 1 



Royal Astjkon. Soc. Vol. LVII. 



.»:> 







Wmk0&£Wyix 






^pllllll 
















^wafS?W::5a 






■Ills: 




^HsR?;?;::lsss 










J||||p:f:-:©; : :WK : S 












^^^:*;#:%>> : >S: : ::: 



















140 






Mr. Ernest W. Brown, Theory of 1 


he 











Latitude, Coefficients of Sines (continued). 



P.O. 


2. V. 


F. D. 


Ooeffi 


yflV 


3 1 


1 2 


n 
— o'oo3 









- -063 






• .'. ~ z : 


- -056 






-4 


+ 'OOI 






-6 


— 'OOI 


|;; : i;iffii-i:±S; : ffi:':| 


-3 - 1 


1 6 


+ -004 






4 


+ '002 






2 


#■ ;OIO 









+ '024 






— 2 


+ -005 




3 -1 


1 2 


''.•;+.•'; 'cJii, 









+ -076 






— 2 


— '019 




-3 1 


1 6 


— 'OOI 






4 


+ '002 






2 


+ '002 









— "024 






— 2 


— 'OIO 


yeV 2 


2 2 


1 


— 'QOS 






—2 


— '023 






—4 


— -003 






-6 


— "OOI 




— 2 —2 


1 6 


+ 'OOI 






4 


+ '013 






2 


— - 002 


! 







+ - °03 , 


I 


2 — 2 


1 2 


+ "007 


I 







+ -016 


I 




— 2 


+ "013 






-4 


+ "OOI ' 




— 2 2 


1 4 


— -003 1 






2 


: -^ '006 









— •006 






— 2 


— '005 



i P.O. 

1 '■" ■ ' 


I. 


I'. 


F 


D 


Ooeff, j 


; yee' 3 


f 


3 


1 




-r-2 

-4 


— O'OOI i 

. : -'.-I 

— -oio 

— 'OOI 




— I 


-3 


1 


4 



+ 'OOI 

1 
-f "OIT 

+ 'OOI ] 


:-:^ :::y 


I 


-3 


I 





+ '002 

+ 'O02 , 




— I 


3 


I 


0; 

— 2 : 


— '002 i 

— '002 ] 


y'eaj 


I 





3 


* 1 
— 1 

-3 
-5 


+ '002 
+ - O04 1 
+ '002 I 
— 'OOI 




: —I 





3 


3 

1 ■ 
— i . 

-3 


+ 'OOI 

+ "003 

— "OOI 

— "OOI 


yVaj 





1 


3 


1 

— 1 

_3 


— 'OOI 

•000 

+ 'OOI 







— i 


3 


— 1 


— 'OOI 


ye'ai 


3 





1 


1 
— 1 

-3 


— '006 1 

+ '°'3 
+ '003 


: ; yy : ; : ■ : ; 


-3 





1 


5 
3 
1 

— 1 


— -002 
•OOO 1 

— 'ooi; '• 

1 

+ '002 


yeVtq 


'"2 


I 


1 


1 
— 1 
-3 


+ 'OIO 
— 'OOI 

+ '°°3 I 



■ 



in 

m 

m 



m 



!;• 



Mullmt uf tin. Moi'U. 



Latitude. Coefficients of Sines (concluded). 



Ml 



i 



m 



I 



1 

P.O. 


I. 


V. 


P. 


i>. 


Coeff, 


■ye i e'a l 


—■?. 


— I 


I 


3 
i 


— C003 

— '002 

— -003 




2 


— I 


X 


i 
— i 

-3 


— "O02 

— 'o[6 

+ 'OOI 




— 2 


; I 


I 


3 
i 


- "°°3 
+ 'oi7 

+ "OOI 


y 6 e 


I 





5 


o 

— 2 


+ -002 
+ 'OOI 




— I 


o 


5 


O 

— 2 


+ '003 

+ "OOI 













P. c. 


, 


v. 


F. n. 


Coeff. 


1 

y'V 




o 


32 
o 


— O'OOI 

— '014 




-3 





3 4 

2 
O 

— 2 


— -003 

+ '002 
+ 'OOI 

+ 'OOI 


ye 5 


5 





I 2 
O 

— 2 

-4 


+ 'OOI 

+ -oi8 

— '012 
+ 'OOI 




-5 


o 


I 2 

O 

— 2 


+ '002 

— - oo6 

— 'OOI 



jjilfll 



■3 



mm 



WM 



llll 

Wm 
'4BBm 

mm 

llll 



I 



1 



mm 



III! 



*»&i 



■ 
■r 



14^ 



Mr. Ernest W. Brown, Theory of the 



I: 



266. Parallax. Coefficients of Cosines. 



P. c. 


I. V. V. D. 


1 

Coeff. 




P, C. 


I. 1'. F. D. 


. Coeff. 


I 


000 6 


+ 0-0032 




ee' 


1104 


— 0-0012 




4 


+ "2607 






2 


j - -0485 




2 


+ 28-2333 









— ^s 02 







+ 34227000 






— 2 


+ 1 '445 5 


e 


100 6 


+ '0007 






—4 


+ -0674 




4 


+ '0433 




1 


-6 


+ '0015 




2 


+ 3'°86i 




i 


1 —1 6 


+ "OOOI 







+ 186-5398 






4 


4- -0060 




— 2 


+ 34'3"7 






2 


+ '2305 




-4 


+ -6008 









+ 1-1542 




-6 


+ -0086 






— 2 


— -2260 




-8 


+ "0002 






—4 


— -0I02 | 


e< 


010 4 


- -0053 






-6 


— -0005 




2 


— '3OO4 




e n 


0202 


— -O028 ! 







— "4002 









- -0086 1 




— 2 


+ I-9202 






— 2 


+ -0920 1 




— 4 


+ '°339 






— 4 


+ -0028 ' 




-6 


+ -0006 




7 2 


0022 


— -0009 


a, 


000 5 


+ -oooi 









— '0124 ; 




3 


+ '0023 






— 2 


— -1052 




1 


- '9752 






— 4 


+ -0031 


e 2 


200 6 


+ "OOOI 






-6 


+ 'OOOI 




4 


+ -0054 




ea { I 


1003 


— -0003 




2 


+ "2833 






1 


— -1090 







+ 10-1657 j 






— 1 | 


+ -0118 




— 2 


- '3°39 




1 


-3 


- '0385 




— 4 


+ '3722 




I 


-5 


- -0003 




-6 


+ -0109 




e'a, | 


0103 


+ -0027 




-8 


+ -0002 




I 

i 


i [ 

— 1 ! 


+ "1492 
— '0037 


















1 


-3 | 


+ -0007 









Motion of the Moon. 



*43 



■mi 

i 



Parallax. Coefficients o£ 'Cosines (continued). 



■P 



• .* 



, 



Mi 



P.O. 


/. Z'. F. D. 


Coeff. 


e 3 


3004 


1 

+ ©•0007 




2 


+ -0243 




O 


+ '6215 




— 2 


— '1187 




- 4 


+ -0074 




-6 


+ -0046 




-8 


+ "0002 


eV 


2 104 


— ■000 1 




2 


— -0051 







- '1039 




— 2 


— '0192 




-4 


+ '0324 




-6 . 


+ '0017 




2 —1 4 


+ "0007 




2 


+ -0213 







+ "1270 




— 2 


— '0017 




—4 


- '0043 




-6 


— '0002 


ee' 2 


1202 


+ '0001 







— 'oio6 




— 2 


+ '0485 




-4 


+ *oo44 




-6 


+ '0002 




1 —2 4 


+ -0005 




2 


+ '0112 







+ -0196 




— 2 


— "0213 




—4 


— '0003 


e' 3 


0300 


— '0002 




— 2 


4- "0036 




—4 


+ '0002 



P.O. 



I'. F. 



ey' 



ey z 



y 2 a, 



022 
o 

— 2 
-4 
-6 

—2 4 

2 
o 

— 2 

-4 

12 

— 2 

-4 
1—2 4 

2. 
o 

— 2. 
001 

— 1 
-3 

. -S 

1 o 3 

1 

— 1 

—3 

1 o 1 

— 1 

. — 3 

2 o 1 ' 

— r ■ 

o 2 I 

— 1 

-3 



Coeff. 



4-o'oobi 

— 'ooio 

— -0833 
+ '0014 
+ '0002 

— -0005 

— '0481 

— -7136 

— 'Oil 2 

— '0001 

+' "00 1 3 

— - oo66 
+ '0005 

— 'OOOI 

+ '0014 
+ "0017 

+• "OOOI 

— 'OIOO 

+ -0155 

— -0088 

— - ooo8 
+ -0003 
+ "0164 

•0000 

— -0025 

— "0014 
•0000 

+ -0036 

— -0003 
+ '0003 

+ 'OOOI 

:+ 'OOOI 

+ "0071 

— '0017 





- : - : **' r iiis§ 
•:p 

sill 
' ; i 



'44 



Mr. Ernest 'W. Brown, Theory of the 



Parallax. Coefficients of Cosines -(continued). 



P. 0. 


I. 1'. F. I). 

1 


i Goeff. 

1 

; |:;:; : ;;;x;:::;::;:;:;;:;:; ; ; : ;::;:-; : ;xi::: } 




P.O. 


I. 1'. 


F. D. 


[ Ooeff. 


e* 


I 
14004 


-| 

+ 0'OOOI 


g2y2 


2 


. ' 2. z '• 


" 

— O'OOOI 




2 


+ - ooi8 


1 









+ '0004 







+ -0401 


1 ■' • • ; 






'.. —2. 


— '0090 




■'..'— 2 ' •. 


- -0x30 


t' 






-4 


+ '0002 




-4 


+ 'OOOI 


1 




2 


— 2 2 


— "0053 




-6 


+ 'OO02 


1 









+ '0004 


i 


-8 


+ "OOOI 


I' 






• ■ . — 3 • 


— '0141 


eV 


3102 


— "0006 


1 ■'■•• ' 






-4 


— '0004 







— '0097 




ee'y % 


1 1 


2 


+ 'OOOI 




™-:2' 


- '°°45 








— 2 


— "0032 




—4 


+ "0006 






| 


_ 4 


+ 'OOOI 




-6 


+ "0005 






1 1 


— 2 ' 2'. 


+ '0006 




-8 


+ 'OOOI 








; ° ■ 


+ '0024 




3—10 4 


+ "OOOI 








— 2 


— '0006 




2 


-f -0017 






1 —1 


: 2" ■ * : 2 


— 'OOOI 







4- '0115 








O 


+ '0003 




— 2 


— -0017 








.'' —2 


+ '0004 




-4 


+ '0002 






1 —1 


~2 4 


— 'OOOI 




-6 


— "OOOI 








2 


— -0027 


eV 2 


2200 


— '0009 








° ! 


— '0029 




. '■■-rz .-•'• 


— -0009 




e'iyi 


2 


2 — 2 


— -0004 




—'4 


+ "002O 




y* 





4 ' 


•0000 




-6 


+ ; •OOOI 




e 3 a l 


3 


1 


— '0009 




2 — 2 4 


+ 'OOOI 








—1 


+ '0017 




2 


+ "0013 




1 




-3 


+ "OOOI 







+ '0024 








—5 ! — '00O2 




— 2 


— 'OOOI 




e i e'a l 


2 1 


3 1 + ;bbp2 




-4 


— -0005 








1 + '0015 


ee' s 


1300 

—2 
—4 


— '0002 

+ '0014 
+ '0002 








— I — '0O02 

— 3 | — '0005 
— 5 i — '0002 


j 


1-30 a 


+ '0004 






2 "—1. 


1 I •'. "^- 'PP°5 ■ 




..0 :•'.: 


+ '0004 








— 1 | — '0028 


1 


• .~* 2 


+ "0002 








-3 


— '0005 










-+■ -0002 



m 
m 



m 
m 



m 

Si 
■ 



m 

pi 



mmlmm 






i$ 



m 



is. 

Wm 







Motion of the Moon. 145 





Parallax. Coefficients of Cosines (concluded). 



P.C. 


I. 


V. 


F, 


I). 


Coeff. 


t 


I • 





2 


. 1 : 
— I- 

-3 

-5 


+ 0"Q002 

4- "ooio 

+ "0002 

. '1 
— "O0O2 


! 


I 





— 2 


3 
1 

— 1 
-3 


— ' "0002 
•OOOO 

+ '0006 
+ "0004 


e'y i a l 





1 


2 


— 1 



+ 'OOOI 1 

I 

— "000 T 







1 


— 2 


3 


: 

+ "OOOI 1 




: : : 






i 


— -0003 

:1 



P.C. 


I. 


V. 


F. D. 


Coeff. 


e* 


5 





2 


, : ~2 : 
— 4 


+ 0"0002 
+ "0026 

— "0012 
+ "OOOI 


e 3 y 2 


3 





2 2 
O 

— 2 

—4 


— 'OOOI 
"OOOO 

— "OO09 
+ "OOOI 




3 





— 2 2 


; —2 

-4 


— -0005 

— -0003 

+ 'OOOI 

— "0008 


«y 4 


1 


0^ 


—4 ' • • 2 ■ 
0. . 


+ "0002 
— "OOOI 



Ha.vcrford College. : 
1 904 December 3. 



Errata (additional to those given on p. 202 in Part II.) 

Part I. (vol. liii.), p. 43, line 12, for "equal masses " read " masses equal to their actual masses." 
„ „ 46, „ 18, for "parallactic inequality" read "the principal parallactic inequality 

in longitude." 
si si 83, „ 2 (in some copies), for " v r _ c , _ ( _ l " read " i',_,i j-w-" 

Part II. (vol. liii.), p. 166, line 21, for the denominator " «/%' " read " u'^." 

Part III. (vol. liv.), p. 5, line 6 from bottom, for " K % " read " Il x ." 

„ ,, 19, „ 6 „ for "+"ooooi o" read " ■—••ooooo 9." 

,. ,. „ 4 „ for " —"00019 2 " rea d " —'00012 9." 

„ „ last line, for " +-19822 " read " +-19828." 



Theory of the Motion of the Moon; containing a New Calculation of the Expressions 
for the Coordinates of the Moon in Terms of the Time. By Ernest W. Brown, 
M.A., Se.D., F.R.S. 

[Received April 13; read May 8, 1908.] 



PART V. CHAPTERS X.-XV. 



The previous parts of this Memoir have been published in the Memoirs of the Royal 
Astronomical Society under the same title in 1897, 1899, 1900, 1905. They con- 
tained the solution of what I have called the main problem — the motion of the 
Moon under the attraction of the Earth and the Sun. which are supposed to move 
round one another in a fixed elliptic orbit in the plane of reference, the three bodies 
being treated as spheres. This fifth and last part treats of the effects of all other 
gravitational causes — the direct and indirect attractions of the planets, the deviations 
of the masses of the Earth and Moon from mechanical sphericity, and various 
minor perturbations which had been specifically excluded. 

The problem presented by these additional perturbing forces is a complicated one. 
In order to clear the ground, it was first assumed that each force contained a small 
factor whose square could be neglected, so that the perturbations could be separately 
considered ; then the perturbations were supposed to be expressible by a series of 
secular and periodic terms, each of which, under the same assumption, could also 
be separately considered. Chap. X. therefore contains the method for finding the 
effect of a perturbation expressed by a periodic or secular term, and in Chaps. XL, 
XII, XIII. this method is successively applied to direct and to indirect planetary action 
and to the action of the figures of the Earth and Moon. In Chap. XIV., in addition to 
minor perturbations, the effects of including the squares and higher powers of the small 
factor is considered : a few very small terms were found. In Chap. XV. the results 
from all these perturbations are gathered together, so that the final expressions for 
the coordinates of the Moon in terms of the time are obtained by adding the results in 
Chap. XV. to those previously given in Chap. IX. 

EoYAL ASTRON. SoC, VOL. LIX. I 



p^^^^^^fp^^^^l^^^^p^lplp^^^p^^pppl 



2 Mr. Ernest W. Brown, Theory of the 

The whole question of these perturbations, in spite of this division into parts, 
is still complex, and, in preparing the results for publication, the choice of the portions 
to be presented in detail was an embarrassing one. It was made more so by the 
conditions which rendered necessary a previous publication of the work for the 
direct planetary inequalities in separate form.* This work included a full investigation 
of the equations of variations which are needed for inequalities arising from all 
sources, and there was thus a choice between repeating this investigation and leaving 
the present Memoir incomplete. The same difficulty arose with the direct inequalities 
and with some other subjects which I have discussed in previous papers. I finally 
adopted the plan of inserting this previous work either when it was essential for clear 
presentation, or when it was sufficiently brief to occupy but little space, or when the 
proofs could be considerably improved. Thus, in Chap. X., the equations of 
variations, the idea of which is due to G. W. Hill, are rapidly put into the required 
form (A.P.E., Sect. I.); the formulas for obtaining derivatives with respect to n 
from a theory in which the numerical value of n'jn has been substituted [Trans. Amer. 
Math. Soc, vol. iv. ) are deduced in a few lines; the methods for dealing immediately 
with non-periodic terms {Proc Lond. Math. Soc, vol. xxviii., and Trans. Amer. Math. 
Soc, vol. v.) are partly developed, but the results (ib., and Monthly Notices, vol. Ivii.) are 
only quoted. In Chap. XL the proof of the theorem on which the method for the direct 
inequalities is based (A.P.E., Sect. II.) is exhibited in a simple form, but the full 
algebraical results are merely quoted ; for the rest of this part of the subject the 
methods are described in general terms and the final results alone given. In Chap. 
XII. a theorem for finding quickly the effects of long-period and secular inequalities 
in the Earth's motion is simply quoted {Trans. Amer. Math. Soc, vol. vi.) ; this 
theorem was also used several times in Chap. XI v. ; but the brief derivation of the 
disturbing function for the motion of the ecliptic {Monthly Notices, vol. Ixviii.) is 
given in full. 

It was found to be impracticable to present within reasonable limits' much of 
the work actually performed. Numerous rough computations were made to find out 
whether coefficients or classes of coefficients were sensible ; when they were insensible, 
a simple note, often the result of clays or weeks of work, is made to that effect, but 
the organised plan of procedure always used in such cases is generally described ; this 
is the case with most of the results in Chap. XIV. 

Owing to the indications furnished by observation of an inequality or inequalities 
with a coefficient or coefficients of the order of 10" of arc and of very long period, one 
of the chief objects in view has been an investigation of such terms, and the " sieves" 
used in Chaps. XL, XII., XIV. were devised for this purpose. No large coefficients 
beyond those already known have been found. Moreover, the search has led more 
and more, to the conclusion that no such terms can possibly arise with the laws of 
motion and of gravitation on which this theoretical investigation is based. If these 
* Adams Prize Essay, Pitt Press, Cambridge, 1908. This will be referred to below by the letters A. P.M. 



Motion of the Moon. 



3 



inequalities have a real existence, it would seem that the cause must be sought in some 
action not purely gravitational. 

No part of the numerical work, except some of the multiplications of series which 
were necessary to find the derivatives with respect to n in Chap. X., Sect, (iii), has been 
turned over to computers. There are so many delicate points to consider, and so many 
terms and classes of terms have special peculiarities which permit the calculations to be 
much abbreviated, that to obtain the accuracy at which I have aimed by a general plan 
which could not permit these peculiarities to be used, would have involved an amount 
of computation out of all proportion to the final results. In fact, not more than 
one-third of the time occupied by these investigations has been spent on accurate 
numerical work. But all such work has been gone over at least twice, in many cases 
three times, and tested by comparisons and various methods, whenever possible. 

This part concludes the theoretical investigation of the motion of the Moon under 
the attraction of gravitation. Its natural sequence — the formation of tables to facilitate 
the accurate computation of the position of the Moon at any time or for the purposes 
of an ephemeris — has already been arranged for, and will be undertaken at an early 
date. But here also it seems advisable not to set the computers at work until an 
extended examination of methods which will best serve the purpose, and of the 
properties of the final results, has been made, so that the highest possible accuracy may 
be obtained within the limits set by practical necessities. 

The table of contents of Part V. follows. 

Chapter X. — Methods for finding the remaining Lunar Perturbations. 

Section (i). The equations of variations. 

Section (ii). Reduction of the equations to numerical form. 

Section (iii). Derivatives with respect to n. 

Section (iv). The final form of the equations of variations. 

Section (v). Numerical values of functions of the lunar coordinates. 



Chapter XL — The Direct Action of the, Planets. 

Section (i). The disturbing function. 

Section (ii). The computation of the coefficients P ; . 

Section (iii). The sieve. 

Section (iv). Numerical values of the elements. 

Section (v). The final results for the direct action. 

Chapter XII. — The Indirect Action of the Planets. 

Section (i). The disturbing function. 

Section (ii). The computation otSp', SV. 

Section (iii). Second method. Application to non-periodic changes. 

Section (iv). The motion of the ecliptic. 



Mr, Ernest TV', Brown, Theory of the 

Section (v). Numerical values of the Earth's perturbations. 

Section (vi). A sieve for the rejection of insensible coefficients. 

Section (vii). Computation of the lunar perturbations. 

Section (viii). Final results for the indirect action. 

Chapter XIII. — Action of the Figures of the Earth and Moon. 

Section (i). The disturbing function for the figure of the Earth. 

Section (ii). Numerical results. 

Section (iii). The action of the figure of the Moon. 

Chapter XIV. — -The Remaining Perturbations. 

Section (i). Corrections due to the masses of the Earth and Moon. 

Section (ii). The terms of the second and higher orders. 

Section (iii). Calculation of the terms. 

Section (iv). Perturbations with unknown constants. 

Chapter XV. — The Final Expressions for the Moon's Coordinates. 



Motion of the Moon. 



CHAPTER X. 



METHODS FOR FINDING THE REMAINING LUNAR PERTURBATIONS. 



Section (i). The Equations of Variations. 



267. The Canonical Equations. — The problem solved in the preceding chapters 
may be stated as a solution in series of the equations 



dx, 
dt 



dll 

"dfd 



dy 



,711 



ay,- fin TT , , ., .-> ,->> ,-, 

-^ = - , 11 = Uy 2 + v., 2 + Y.. 2 ) - 1 

dt (fa, ' 2 x " l ' 2 " 3 ; 



(1), 



where x, (i = 1, 2, 3) are the coordinates of the Moon referred to fixed axes, and F is 
the force function under the hypotheses stated in Chap. I., Sect. (i). 

Let w lt iv 1 — w 2 , iv 1 — iVg be the Moon's mean longitude, and the arguments of the 
principal elliptic term, and of the principal term in latitude, respectively, so that w. 2 , iv 3 
are the "mean longitudes of the perigee and node'' resulting from the solution of (1). 
Then Jacobi's method shows that, if a quantity R, be added to F, and if it be expressed 
in terms of iv v u> 2 , w 3 and c v c 2 , c 3 , the other three arbitraries of the solution of (1), 
the latter three may be so chosen that the solution of the differential equations 

dc, dll ,hv : dR 



dt dw. 



dt 



dc, 



+ h 



(2) 



will give variable values of the c { . w h which, when substituted in the expressions for 
the Moon's coordinates and velocities instead of their former values, will give the 
Moon's position and motion under the force function F+li. Here b 1} 6 2 , b s are the 
coefficients of t in the angles iv V/ w 2 , iv s . Hence, when .R = o, b 1 = n, the mean motion, 
and b 2 , b s are the mean motions of the perigee and node. They are functions of c x , c 2 , c 3 
and the constants present in the differential equations; and b^dc^ + b^dcy + b^lc^ is a 
perfect differential. 

268. Transformation to the Variables n, c 2 , c s , u\— -Change to the system n. c.-,, c v 
retaining the w ; unchanged, so that c 1 is now a function of n, c 2 , c 3 . Then 



Adn dc, 



dc, 



- l dc„ ) + b,dc„ + i)„dc, 
de 3 -J - A " ° 

is a perfect differential, and therefore, since b Y = n, 



d h - 

dc„ 



dA\ 
dn 



dc x 
dc. 



dn 



db 9 _ db B 
dc, dc 



wssmifBiam 



6 Mr. Ernest W. Brown, Theory of the 

The equations (2) then become 



dn 



dR db a dR db,, dR 



dt a 2 fj\ du\ dn dw 2 dn di, 



dc 2 _dR 
W~dv„' 



dc $ _dR 



dt 



dm.. 



dw 1 _ 1 dR , 



dw„ 
It 



where 



dc„ ■' \ dt 



dn 



h \db 2 

Viin ■■ 



div„ 



clR 



+ b„ 



■dio 1 
lit' 



,^8 

dn 



(3), 



-a?(i , 



and all the functions are supposed to be expressed in terms of n, c 2 , c 8 , w t . It is to be 
remembered throughout that, when using the variables w { instead of the constant parts 
of those angles, the derivatives of w t with respect to n, c 2 , c 8 (or any functions of them) 
are zero. 

269. Solution of the Equations.— It is supposed that R contains a small factor 
whose square may be neglected, and consequently that we may substitute the undis- 
turbed values of n, c 2 , c 3 , w { in R, that is, the values furnished by the solution of the 
main problem. In all the cases to be considered under this method, R can then be 
expressed as the sum of non-periodic and periodic terms, and each of these terms may 
be separately treated. First, for periodic terms, put 

R = A' cos (qt + q') = A' cos (i 1 w 1 + i 2 w 2 + i s w 3 + q"t + q") , 

where q", q'" are constants independent of n, c 2 , c 8 , w it and A' is a function of n, c 2 , c 8 
only. Substituting in the first three of equations (3) and integrating, we obtain 



A' dq 
a 2 /Jq dn 



cos (qt + q') , 8a, ■■ 



i,A' 



i„A' 



•kt + q'), Sc :i = ^± COS (qt + q') , 



(4), 



where Sn, h 2 , Sc 9 are the additions to n, c 2 , c 3 due to R. The arbitrary constants due 
to integration are given zero values. 

If Sb v §b 2 , Sb 8 are the corresponding changes in \ = n, b 2 , b n due to these changes 
in n, c 2 , c 8 , we have <f6 x = fe, and from (2a) and (4) 

%h - dl> ^r,4- db ^ l - -u db 2S„ I 1 dq db 2 , dq\A' . t ,, 

B h = ^Sn + §S8 02 + p S( ^(^pp + ^ co 

dn de, 2 dr 3 \ a 2 f3 dn dn dcj q u J ' 

Denoting by Sw t the additional part of w { , substituting the value of R and these 
results in the second three of equations (3), and integrating, we obtain 



s 1 / 1 dA' A' dq\ . . , , x 



s W2 = { ('_L ^^_2£V +3 _ j_ d( i db M' I sin (af+r/) 

2 S W/? dn dn do J q + \dc 2 a?ji dn dnJJ* } " m W ' + q) ' 



8w„ 



1 dA' db B dA'\i f dq i dq db,\A'\ . , , ,. 



(5). 



Motion of the Moon. 



7 



The equations (4), (5) constitute the theoretical solution of the problem. If X be 
one of the Moon's coordinates, then the additional terms due to R are given by 

5,. d\« , dk« , d\„ , rfA.„ dk „ <lk ^ 

bk = ~--bn + — M:, + -—bc„ + , bw, + —— bw 9 + bw. . 

dn dc 2 ~ dc B aw 1 dw 2 " dw & 

2 jo. The Constant Term of R. — Denoting it by R , we have, instead of the 
equations (4), Sn, Sc 2 , Sc s constant. These constants are at our disposal. Put Sn = S Q n, 
Sc» = o, Sc s = o. Then, instead of equations (5) we have 

dR n 



Sw 1 = 



a 2 /3dn ° 



bw 9 - 



f dR a db.)~ 



dc„ 



dn 



6 V dc, dn V 



where the additive constants are made zero. 

Now, since the mean longitude is a quantity observed directly, we so choose § n that 
■w\ is still represented by nt + e, and therefore Siv 1 = o. Whence the changes in the 
angles iv 2 , w 3 are obtained by adding to their motions the quantities 



8/;.,= 



dR^ 

dc,. 



which include the change S n. 
Since Sw 1 = o, we have 



»0« = 



bb s = 



'//.' 



dR 

dc„ 



(6). 



(7). 



This change in n must be substituted only in the coefficients of the periodic terms 
representing the Moon's coordinates, amongst them, the principal elliptic term in 
longitude (2c with sufficient accuracy) and the principal term, in latitude (27). These, 
again, are quantities observed directly, and therefore, as we wish to retain the same 
expressions to denote these coefficients, it is necessary to add to e, 7 in all other terms 
the amounts 



8 e = 



de „ 



respectively. It is true that these produce further changes in b 2 , b s , but they are quite 
insensible. Indeed, the changes (8) produce alterations less than c/ /- oi in any coefficient. 
The changes, as found from the methods of Sect, (ii) below, give 

V= - [3745 2 ] V . 8 y= - [3-88t2]S m . 
If R contains a non-periodic term of the form R p t p , where R p is independent of t, 
the corresponding changes in Sn, Sc 2 , Sc s are zero, and 



biv =— dE " f " +l 
1 o?(i dn p + 1 



Btv„ 



dR„ jp +1 ,db ?8w 
dc 2 p + 1 dn 1 ' 



8w 3 = 



dR„ t" +1 , db, s 
dc s p + 1 dn 



Si 
pi 1 

1 



Section (ii). Redaction of the Equations to Numerical For 



m. 



271. Computation of the c { . — The coordinates x,- of the Moon have been expressed 
in terms of e, k instead of c 2 , c 3 ; we must find the relations between the two sets of 
constants. 



8 Mr, Eksest W. Brown, Theory of the 

Owing to the canonical forms of (i), (2), it is well known that 2,( yM + u Y lc { ) = dS, 
a perfect differential. Hence if every quantity be expressed in terms of c i} w if 

^=S,yA f^ + ^% (*,; = !, 2, 3). 

dw< dwi dCi act 

But since T = dxjdt, these equations show that dS/dw { consists only of cosines and 
dSldct-Wt of smes of the angles present in the solution of the mam problem and 
therefore that 5-2^ + periodic terms. Denoting by [<?] the constant term of the 
expansion of Q as a sum of periodic and non-periodic terms, and substituting the 
value of S in the first of the previous equations, we obtain 

' dXj dXjl ...... (9). 

1 { ' dt died* 
the equations for finding c ( > 

The co-ordinates x x , x 2 , x 3 are here those referred to fixed axes. * e therefore put 

Xl + x 2 i - « exp. i(n'* + , x a - x 2 t = 8 exp. - .(»'* + , x 3 = a , 

to reduce to our earlier notation. _ 

Next, lt , s have been expanded in positive and negative powers oi I , I , i , { , 

Wh6re ii-erp.iK-»'*-0, «» = « P.^,--,). P = exp. .(«,-«-,), «--exp.^ + «), 

and numerical values have not been substituted in the exponents. In line therefore 
with the former definition of D, I put 

D,=i}~, A-Ki, D^i?±, V**-^' 

so that d 

A = t (z> 1 + A + A) , 4- = - lD - ' ^7 " " ^ ' 

D = D 1 + cD c + g-Dg + mfl m . 

Making these substitutions in the expressions for c it and remembering that, as we 
only need the constant term,/(«, s)+f(s ; u) can be written 2f(u, s), we obtain 

Cl = - In - ra' MD + i + m) h' • (A + A + A - J )«' + ^ ' (A + D < + AM) > } 

_3_- = r<z> + 1 + my ■ z>, «' + z> 2 ■ M » -~ 8 v = [(^ + * + m)rt ' ' ^ + Dz ' DA J 

where tt' = 'M^ _1 ) s ' = " s v , , ■ • 

As a matter of fact, Cl will not be needed. The values of c,, c 3 have been given m 

Chap. VII. , § H5 (there called ft, ft). They are 

,^6^e% 2 -"oonoeV 2 , ) 



%= - '11844 440.2- -023246* - ^eaese-k 2 - 'oonoeV 2 > 



e„ 

_a_= - 2 002 

.2 



wa 



.05 9 k*- r 9 6 3 76k 4 - -285 4 6kV - -00568!^ ) 



The forms of these expressions are important : c 2 is divisible by e 2 and c 8 by k 2 , and 
if we neglect powers of m, c 2 = -\na\\ c.= - 2naV. The numerical coefficients are 



mmmmm 



Motion of the Moor 



to be considered as functions of n. It must be noticed, however, that in finding the 
derivative of c 2 with respect to n, the terms in m 2 , m 3 , . . . diminish the value obtained 
from the principal term by nearly one-half ; the derivative of c 3 is not much altered by 
these higher terms.* It is in general true that quantities depending on e, b 2 in any way 
are slowly convergent along powers of m, while those depending on k, b s are rapidly 
convergent. 

272. Derivatives with respect to c 2 , c s . — These may be obtained from the deriva- 
tives with respect to e, k by solving the equations 



dQ dQ dc 2 dQ dc 3 
de dc, 2 de dc z de 



. dQ dQ ,dc 2 dQ dc s 
dk dc 2 dk dc 3 dk 



where Q is any function under consideration. Inserting the numerical values of e, k 
obtained from §§ 192, 193 of Chap. IX., I find. 



adQ , r i dQ r - -,. dQ 

k =+[2S423> *^ [r773l]k "rfk' 



;M 2^= +[2-o929]k^-[77386>^ 

dc, L J yj dk L '•* J de 



In § 145, Chap. VII., will be found the materials to obtain b 2 , h 3 (there denoted by 
ttj, Oj). From them I obtain, with the help of equations (12), 



de„ 



+ [2-3i7S], « 2 -i^= + [2-3960] = a^ 3; fl2 p = _ [ 3. s698 ] . . (I3) . 

273. Derivatives ofb 2 , b 3 with respect to n. — These might have been obtained by 
finding c x from the first of equations (9a) and using the first and second of equations 
(2a). But it was found to be much more simple and sufficiently accurate to use the 
existing literal developments in combination with the numerical developments. The 
method for doing this I have given in a former paper. t It is as follows : — 

Let 

/(m) = a + ttjlll + a 2 m 2 + . . • 

and denote byj^(m) the sum of the first i terms of this series, and by an accent the 
derivative with respect to m. Then 



m/'(na) = m//(m) + i 



i , * + ! >■ 

I * +1 



,1+2 i+o 



If i be not too small, and the series not converging too slowly, the error committed by 
putting 

m/'(m) = m//(m) + i[f(m) -/ 4 (m)] 

will be small compared with the true value of m/ v (m). When the series appears to 
be diminishing with fair regularity, the use of i + proper fraction instead of i in the 
last formula will probably give greater accuracy. As a matter of fact, I have only 
used this to find the greatest possible error which could have been committed, so as 
to avoid any sensible error in the result. 

The derivatives of & 2 , b % are found from the literal \ and numerical values of Hill 
* A.P.E., p. 8. f Monthly Notices, vol. Ivii. p. 346. \ Ann. of Math., vol. ix. p. 40. 

Royal Astron. Soc, Vol. LIX. 2 



IO 



Mr, Ernest V. Brown, Theory of (he 



for the part of b, depending on m only, and of Adams * for the corresponding part of 
h, ; and for the other portions from the literal values of Delaunay + after a test by a 
special method which I have given earlier,} combined with my numerical values. 

These latter portions are expressed in terms of e, 7. Since the derivatives of e/e, 
k/V are insensible to the degree of accuracy required, we can make the change, after 
differentiating Delaunay's series, by the formulae of § 192, Chap. IX. 

The required derivatives with respect to n are obtained from equations (11) 
combined with 

d J* = ('' ( ±\ <!Q('!h\ 'I®(' Ic s) . . ■ (14). 

du \dn ) ~ dcXdn J ' dcS,dn ) 

where the brackets denote that the enclosed functions are expressed in terms of n, e, k. 
The derivatives of c,, c 3 are obtained from Newcomb's transformation § of Delaunay's 
literal values for G-L = c 2 , lf-G = o s , combined with the numerical values of c 2 , c, 
given above. They are only needed to two significant figures, jl 

The other derivatives with respect to n are found in the next section. 

These apparently complicated processes are constructed to avoid the slow 
convergence which occurs with certain of the literal series arranged in powers of m, 
and I believe they have achieved the object in view. 

274. The subsidiary results are 

@^S-4637]«S G&h^-^K, ©)=-[-7o 9 ], ©H[3'5736], 



\dn 
and the final results 

dn l < j ' d n 



[2-1720], _^= +L3 5733J 



Section (iii). Derivatives with respect to n. 

275. The equations for finding n-derivatives with the system n, c 2 , c s , w t .— The 
problem here is to find these derivatives from a theory in which the numerical value of 
n has been substituted^ I shall show that these derivatives may all be made to 
depend on derivatives with respect to the other five elements, and therefore with 
respect to e, k, iv t . 

Write 

v ('V #'_ i, f <ML\^\f f\ :,..,. (16). 

^Adc, dw, dw, dcJ l ' ' '" 



Then, on changing equations (1) from the canonical set x { , y ( - to the canonical set c h vj { 
by a contact transformation, we have the relations 

* Monthly Notices, vol. xxxviii. p. 48. § Amer. Eph. Papers, vol. v., pt. 4 (1894), pp. 201, 202. 

t G.B., vol. lxxiv. pp. 19 et seq, || A.P.E., p. 8. 

% Monthly Notices, vol. lvii. p. 335. «TI E. W, Brown, Trans, Amer. Math, Soc, vol. iv. pp. 234-248. 



Motion of the Moon. 

Denote by brackets derivatives with respect to the system c { , 
for changing to n, c 2 , c 3 , w { are 



IV s 



df _ (df Y*i d f__( d fi\ d £i ( d f 
dn \defhi ' dc 2 Kdc-Jdr^ \dc 2 



df fdf \cfej 



dc s \dcfdc s \dc 



if 



1 1 
The formulas 

. . (18), 



the other derivatives remaining unchanged. 
Write (see § 268) 

J^L= ( d Lj l !ft f£ _^£i l¥\^ ,lc j^(dK + d h fK + d b d f\^ (i f± , ( I9 ); 

dWf 2 \dw 1 dc 2 dw 2 dc 3 die, J ' dn \dw 1 dn <ho i dn dw % ) ' dn 

then {/,/'} transforms into (/,/'), in which n replaces c 1; and W x replaces w v 
It is convenient to change to our former axes by putting 

. , ,. / ,, i\ du ds 

x 1 + ix 2 = uex]i.(nt + e)t, x x - tx 2 = s exp. - (nt + e)i, x s = 2, M i = ^> s i = f ^' 

and the transformed equations which we shall need are 

(u 1 ,u) = o, (,//,s) = o, (u,z)-=o . . . • . (20). 

Now 

du x d du __ d du -y, /du_ dbA _ d du du <lcj 
dn~~dn ~di~di dn ■^- li \div i dn)~dt dn dW x dn ' 

If this be substituted in the first of equations (20), we obtain 



d du du dU jr 2 
dt dn dn dt 



dc x du d du _ du d du du^ d_ du_ jlu_ d du^ 
dn dw 2 dt dc 2 dc 2 dt dw 2 div s dt dc a de 3 dt dn> 3 ' 



du db„ du db„ du 



- + 



- + ■ 



where 

JJ= t - , -- . 

dw x dn div 2 dn dw a 

Dividing by TJ % and integrating, we have the formula for du/dn : 

du 



-=U 



wh 



ere 



-/3« 2 = 



dn ' 



dn J\ U* J 
Q = l(xst four terms of (21), C= arbitrary constant. 



") 



As du/dn can contain no terms proportional to the time, the constant term of the 
function to be integrated must vanish, and therefore 



- I -f- 



de. 







(22), 



dn [3a 2 

which determines /3. The arbitrary constant C is determined from the second of 
equations (20), and the values of db 2 /dn, db 3 /dn have been found in § 273. The value 
of dz/dn is similarly determined from the third of equations (20) when du/dn is 
known. The process of finding the n-derivatives is therefore reduced to an integration. 
276. Computation of the n-derivatives of u 2 , us, zu, and the products of these 
into (a/rf, — The process given in the preceding paragraph is neither simple in theory 
nor easy for computation. But, in the absence of any other method, it had to be 



>..,,..-,.... ......... w.i^.vMW.'K'.wr.V.v.v/. 



i 2 Air. Ernest W, Brown, Theory of the 

adopted. Various plans for abbreviating the work arose during the computations ; I 
shall not go into these details, bur shall only give the main outlines and formulae which 
were used. 

With the use of the notation defined in § 271, the definitions of U, Q will be 
slightly altered. It is convenient also to replace c 2 , c s by - a 2 me 2 ' 2 , - ahiki, so that 
e 2 /e and k 2 /k are positive. We put, then, 



'11 - 


= w£ _1 , f 2 = -« a W> c 8= -«a 2 k 2 2 > 




U-- 


= U r , + U, = (A + D,. + D, + i)u'- ^D,u' - C pD g u , 
1 v l ' ** dn dn 




a= 


,D,.u d T , , ,D,.Du du , Ti , d ,, , n n / 

e 9 de 2 " e 2 ae 2 " ak 2 ~ 


du 
dk 2 2 


<£= 


■^- I - m )' »^-^=^ + C >- ' 





• ( 2 3)- 

The computation of U, <ft is made in the following way. Let U consist of the 
terms of characteristic unity in U, so that U = {I) t + i)u '. Then 



1 1 U-, U-, 

u 2 u* u* 6 r„ 



After finding i/U i by special values or otherwise from the results of § 44, Chap. II., 
and Ui by multiplication of series, and then U* for the few terms needed, multiplication 
of series and additions quickly give 1/U 2 ; the process is simple, since i/U l converges 
rapidly, and only three or four terms are needed. Next, Q is obtained by multiplication 
of series quite like those used in the earlier chapters, some of which could be made use 
of by means of the identity fDf —f'Df= D(ff') — 2/' Df. To obtain the derivatives 
with respect to e s , k 2 from those for e, k, it was nearly always sufficient to multiply 
by e/e 2 , k/k 2 , respectively, as the formulae (12) of § 272 show. Owing to the form 
of Q, it is necessary to use vl as far as terms with characteristics of one order higher 
in e, k than those needed in dujdn, but the results of previous chapters were far more 
than sufficient. Finally, Q/'U' 2 w T as obtained by multiplication of two series and 
thence /3, cp. 

The value of /3 as found here and tested by two other methods * is '32962. 

The second of equations (20) may be written 



du „ ds TT r, , 

dn dn 



du ds ds du du ds ds du 
_dc, div. 2 dc 9 div„ dc„ dtp, dc s div s . 



Denoting by a bar the change — 1 for i, so that S = — U, and using the new 
definition of U, we find 

1 irffi n , a , j\ 1 1 Dm ds ,D,i du , ,, , ds ,. , du . , 

^ (a c + * + ^^=i-^. _-i-.. _ + />,«. Si - 2 -^.- 5? . . (24 ). 

The multiplications of series for -^ were all at hand, and the computation of UU 
was simple ; to determine C, only the constant terms on each side were needed. 

* From the transformation of Delaunay's L = e 1 (§ 273), and from a theorem connecting it with the 
constant term of the parallax (E. W, Beown, Trans. Amer. Math. Soc, vol. iv. p. 247). 



■" ■■-.<! • 



Motion of the Moon. 
To find the derivative of us = u's', the simplest form appeared to be 

wi-(Ms) = /^ - + u% = i(Y u + 7Tl)(<t> + 4> + 2 V) + 1 (V U - /U)(<j, - £) . 
cm <m «m " 



(25). 



since 



'"-'HM-s'M-^M" 



and the product ws was at hand, the constant disappearing, so that all the terms 
were small. For the purposes of verification 



(s'f/+. /£/)(<£ + <£ + 2 0')= j",+ (j! 



^ 



(26) 



was computed : this gives the more important part of d(us)/dn. 
The derivative of u'' 2 is given by 



n£u'°- = 2u UU + U) = «- + G)(/J 1 + D c + IX, + 1 -- ^A, - '^- 3 A \P 
dn \ 



dn 



dn 



(27 



for which the product u' 2 was available. A multiplication of series gives the required 
function. 

The formula for dzjdn is obtained from the third of equations (20). It is 
transformed into 



>i J f = Z(4, (- u ) - f* , z= Oh + A + A - ^ A - -V^ )« , 



dn 



db s , 
dn 



l D,x du_- [ l) K ii *L + /;-.. du _±D t ,iC dz_ 



" <'•? <'«•> 



de.-, 



//k 2 A k dk 



(28), 



the multiplications of series for x being at hand. 
The derivative of u'z is therefore given by 



n*Hu'z) = - %< + (* + C)f A + IX + 11, + t - ^A - ^ aV^ 
dn ' U A \ 'te dn 7 



( 2 9)> 



which is computed as before. Only the principal terms were required. 

The derivative of z l is given by (28), if we replace 2 by z 2 in the formulae. 

The few terms for derivatives of third-degree products of u, s, z were computed 
from these results. The numerical values are given in Sect, (v), 

277. To find the derivatives of the products of u 2 , us, uz, with (n'rf, we only 
further need the derivative of the last. For the only term which is sensible we 
can use Delatjnay's results, combined with the numerical results of § 266, Chap. IX. 
This is because the constant term of a/r contains no portion depending on e 2 , y 2 , 
and the portions depending on m converge very rapidly ; the other terms in the 
products are of orders <r, y 2 , ra\ e 2 m 2 , nvy 2 , or of higher orders. The results from 
all but those of orders e 2 , y 2 , are practically insensible. 



^^^^^^^^^^^p^tpp^^^^^ip^^^^^^^ 



14 



Mr. Ernest W. Brown, Theory of the 



Section (iv). The Final Form of the Equations of Variations. 

278. Numerical values required. — I gather together the numerical results obtained 
in the preceding sections, so far as they are needed in the equations of variations. 
They are as follows : all numbers whose common logarithms are set down being 
enclosed in square brackets : — 

e= +[27396], e= +[1-0396], y=+[i-652i], k= +[2-6511], ] 

/3= +[7-51801], w = M = +[2-87391], m=- -- ,= +[2-90768], \= +[1-99921] I ■°" J ' : 

11 n-11 a- J 



L^= -[2-1720], a? ^2= +[2-3175], a 2 Li?= +[2-3960], J 

Lj= +[3-5733], a2|*8 =+ [l- 39 6o], a? (] ^== -[3-5698] ! 

„dA' , r . -, dA' r- -,,dA' n dA' r -.dA' r - „,-, dA' 

-na-- = T [ 2 5423^ -[r 773 .]k^ -^ = +[2-o 9a9 ]k_ - [t 73 86]e_ 



(30, 



(32). 



279. The Equations of Variations. — The equations are now put into a form which 
is to admit of direct application to any term of the disturbing function. The 
coefficient of each cosine or sine which constitutes the variation of an element is to be 
expressed in seconds of arc. Put 



j-x I III to 9 . ■ / . 4 . // , f. I. Tib ft) n a / 1 > -■ 

Jtt = — - , n ^& J A 00s (1-.W-, + im'„ + t„w a + t + a ) = -n i & i A cos (qt + ci ) . 

4 m ' A L 1 1 ■ 6 6 1 j- 1 4 m > 

s' = number of seconds of arc in the daily mean motion of the Sun = 3548"'i9 , 

s = ,, ,, ,, ,, argument qt + q ; 

s'2 



i m a- s 



1 m a- tos 



Ql m 



/=— ^ — — 206265= +[ I 2-2935 8 ]— , /=————-= + [7-61748]^, 



4 m a 2 (3 



4 m' a 2 /3 



,, fA n f'A . . n d , , „> . , dA . , , dA 
=• — , G 1 =■>— , , }l A = - — (/la 2 , j,i = e T , ^1 = k - - 



a 2 (fo 



Aj = - i x + , oi486« 2 - -003744*3 , 

A,, = + -0148611 + -006624^ + -ooSadoig, 
A,, = - -003744*1 + - oo826oj 2 - -001238% 

/*,== -[2-i72o],/ 1 + [2-o6o3],7 2 -[i-29n]i 3 , \ 
/*:>,= +[3'57.^3]ii-[i" 2 S 66 ]i2 + [ I ' 6l °9]i3 i 



- = AjOj cos (gtf + q) , 



then the equations of variations are 

8n 

•a 
8c, 
na 

na- 



Sti'j = (AjC+ZjCj) sin (gf + q") . 



2 2 — + [1 '5 1 801 JigC, cos (qt + g') , Sw 2 = (K,G + /.taCj) sin (qt + g') , 
-= + [i-5i8oi]» 3 C7 1 cos (qt + q) , Sjo 8 = (A s C+/x 3 C' 1 ) sin (</i + </') 



If it be desired to find §e, §y, they can be obtained from 



<33>; 



(34); 



\3Si ; 



(36), 



(37;: 



(38). 



te= -[1 2823]-^ + [3 43b ma a + [3-74S2] - > «>T= -L746o]^-[r22 32 ]— , + [ 3 -88i2]- (39). 



S:..' 



Motion of the Moon. i 5 

These formulae are not easily comparable with those of Hill,* EADAU,t or Newcomb,^ 
because I use derivatives with respect to n on the assumption that A' is expressed in 
terms of n, c,, c s , while they suppose that the coefficient is expressed in terms of n, e, 7. 
The chief gain here is the avoidance of the doubtful derivative dc 2 /dn, which in my 
method is certain to the degree of accuracy required ; and further, the method in 
Sect, (iii) of finding the n-derivatives is more simple with the system n, c 2 , c 3 than 
with the other system. The derivative on which nearly all coefficients depend is 

dc s 
d.n 



= —f3a 2 , and this is found accurately. The comparison, however, can be made for 



all terms except those in Sw t involving C t : this has been partly done.§ 

280. Method of using the equations and abbreviations. — The numbers f f are 
the same for all the direct perturbations of a given planet ; for all indirect perturba- 
tions ; they have two values for terms dependent on the motion of the ecliptic; and 
three values for terms depending on the figure of the Earth. The numbers \, X 2) X 3 , 
«.», //.;, are the same for a particular Moon argument in a given class of perturbations ; 
the first and third terms of n 2 and the first and second terms of m 3 can nearly always 
be neglected. Only two numbers, C, G x , have to be computed for each coefficient 
required, and one of these is generally very small. 

The most important particular cases are : — 

(a) i x = L — u = o. Then Sn = Sc 2 = Sc 3 = o and X,-C= o in the Sw { . 

(?;) Terms of long period in which %=|=o. The portions depending on C x are very 
small. 

(e) Terms oj short period approximating to a month or less. The portions 
depending on C are usually small compared with those depending on C lt and it is 
rarely necessary to compute Sc 2 , Sc 3 , owing to the theorem in the next section ; Sn is 
very small. 

(d) Terms depending on iv 2 or w 3 , but not on n\. These are much the most 
troublesome to compute, because Sc 2 , Sw u Sw 2 , or Sc s , Sw 1} Siv s , may produce terms of the 
same order in the Moon's coordinates. But even here Sn is usually insensible, and the 
theorem given in the next paragraph is almost exactly satisfied, so that the computa- 
tions reduce to finding Su^, Syj, (and Sc. 2 as a test only), or Sio t , Sw s (and Sc 3 as a test). 

281. Substitution of the elements in the coordinates. — We have, for the longitude, 



8 V 



«, + -,-■ - 1 )bw, + - bw + -- 61," 
\iiWj J dw 2 ' div 3 



dV v dV, dV„ dV.. 
<•„ + — dn + -— <V.', + ^6 
an dc, " tic. 



(4°) 



om 



Ihe terms arising from the first in this expression, Siv x , are called primary ; those fr 
the others, that is, from the periodic terms, secondary. || All terms in latitude are 
secondary, and there are scarcely any sensible terms in paralb 



ax. 



Aafr. Pap?,-*, vol. iii,, ami Coll. Worls, vol. ii. p. 



Ami. 01,*. Pari,!, vol. xxi. 
Cow, tie Inst. Pub/., Xo. 72. 



Ji A.P.E., p. 10; Monlhly Notice*, vol. Ixviii. p. 167. 
|| A.P.I-;., p. 37. 



■■ - ■ *-*■ ■**■*»* * ■ * • - 9 



1 6 Mr, Ernest W. Brown. Theory of the 

Since Sn is rarely sensible, it is sufficient to obtain dV/dn from Delaunay's * final 
results. The values of dVjde,, dV/dc s are rarely required, except for the purposes of a 
test, owina; to the folio wins; 

282. Theorem.** — If Sw % he confined to the term fi 2 C x sin (qt + q r ) and a coordinate 
X to the term psin (i z 'w 2 + ^- r ), where \f/ is independent ofiv. 2 and ij has the same sign 
as i 2 , and i 2 =j=o, * 2 =f=o, then the variation of the coordinate X due to §n\, §c 2 is 

dk o dX /-v-/-f,,/ •/. ,/i 

— bf 9 + — -dw, = [k„G,fi n s,m\qt + q -i„ w 2 - </>■ } . 
dc 9 aiO:, " ' "' 

That this theorem is true when we neglect all but the lowest power of e present in 
the coefficients and when Sn is negligible is immediately seen. For, in such cases, 
/ 2 = |i 2 |, Sc 2 depends on dE/dw 2 , and Sw 2 on dR/dc 2 . But it appears to hold even more 
accurately. An exactly similar theorem holds for Sc 3 , $iv s , and the terms depending on 
w 3 in a coordinate. In the case of the principal term in latitude due to the figure of 
the Earth it holds within one-tenth of one per cent, of the whole. 

Section (v). Numerical Values of Functions of the Lunar Coordinates. 

283. These functions are chiefly products of the second order in u, s, z. The 

computation of them by multiplications of series is quite like those necessary in 

previous chapters, and indeed the great majority of them had been obtained in 

the solution of the main problem. The formation of the derivatives with respect to 

n, e, k has been developed in previous sections. It is understood that the n 

derivatives are formed with 

j, G . char. = — — - (Ca 2 . char.) , 
&' an' 

where C is any one of these coefficients (given a numerical value here) expressed 
as a function of n, c 2 , c s , except in the case of the figure of the Earth terms which 
are formed withj^iC. char. =d(Cn z . ch&r.)/dn, and the ease of terms due to the motion 
of the ecliptic which are formed with j x s?C . char. = d(na?C. chav.) /dn. The few 
coefficients where an exact computation of these latter forms was necessary are 
given ; in general, these derivatives were not needed, or their values could be set 
down from the known principal term in the literal expansion of the coefficient. 

The values of (a/rf, (a/rf needed for the figure of the Earth terms, are obtained 
directly from the numerical values of ajr in § 266, Chap. IX. These are simple to 
compute for the terms needed, owing to the rapid convergence of the parallax terms. 
The formation of the derivatives has been explained in § 277. 

284. The values of V, U needed to find SV, §U are obtained directly from the 
results of Chap. IX., after division of the coefficients there given by 206265. The 
fact that Sn is always small, combined with the theorem of § 282, and the expressions 
in §279 of Se, Sj in terms of §c. 2 , §c s , make the computation of the derivatives of V 

* Mem. de I'Acad, d. Sc, vol. xxix. chap. 9. t A.P.E., p. 15. 



Motion of the Moon. 



17 



with respect to e, y unnecessary ; they can, however, be found from the results of 
Sect, (v), Chap. IX.. if needed. The value of dV/dn so far as it is wanted is 



oiveii by 



n d :L^ +[2-0465] sin I- [2-4669] siu (2D - Z) - [2-3965] sin 2D 



dn 



(40- 



285. The coefficients M { .— These are defined by the following equations, in which 
d is one of the angles and M t the corresponding coefficient in the expansions of the 
functions set down. Summations for all such terms constitute the complete 
values of the functions. 

According to the previous notation, V, V are the ecliptic true longitudes of the 
Moon and Sun, r 2 = us + z 2 = p' 2 4- z 2 , T = the mean longitude of the Sun, etc. Then 



r' 2 a 2 



„ a" 2 p 2 eos ,„ T7 , N if, cos fl a 2 zp cos.,, , »\ M i sin . 



A/o cos 



a's p(r-' - 52 s ) cos p _ yl) M, cos ^ 1 ct* p 3 cos ( F _ r) = f 8 < ;os 6 
___ -5 ■ sin ^ r ; M.sin ' , r '3 a s sin - 51 ; i¥ 9 sm 



sin 



a'* 2(V 2 - { 

,.'3 n 3 



COS 



™(V-h") = 

1 It V ' 



M w sin e 
- J-/ 10 cos ' 



a^zp 2 cos F _ r iji a sin tf 

.'a ..a tin \ ; - i¥,„ COS 



-'3 .,3 ' sill 



For computation purposes these transform into 



a - us — 2Z- 
r' 2 



- u l\ e T /' „.'■! „2 2\ ->- 3/ ,'2 a -2 



cP u(us ~4z 2 ) e (T-vy = i( a/ 6 + 31 7 ) e ±^ , A . a '! "V 1 '- v "> = i(i¥ 8 ± Af 9 )e±«' 



3 '' ° a 

a,' 2 IZ ■ (US - %Z 2 ) ,ir ,..>;. ,r . fl . a' 3 12 -it 2 



,., a - 10 ' r' 3 a 3 

No values for M m M 12 are given below, as there were no terms large enough to make 
their accurate computation necessary. The computations of these coefficients from the 
results given are quite simple. Besides the operations already mentioned, there are 
multiplications by « /2 /V 2 , (a /2 /V 2 ) cos 2 V, etc., but the simplicity and brevity of these 
processes make further detail unnecessary. 

286. The coefficients marked with a dagger (t) in the following tables include 
characteristics of two orders higher than the principal characteristic, and are therefore 
fully accurate to four significant figures. AH other coefficients were sufficiently accurate 
with the part depending only on the principal characteristic. 



Royal Astron, Soc, Vol. LIX. 



i8 



Mi 



Ebxest W. Brown, Theory of the 



287. Coefficient of C +u x characteristic = coef. of ^"" a ' : x char., in 











a 2 




Char. 





' 


= 


t=i 


j= - 1 


1 





+ 


99262")" 


- '00701! 


- '00701! 


e' 


m 


~r 


O0787f 


4- •00410+ 


- '02992! 


e' 2 


2111 


+ 


0101 t 


+ '0005 t 


"'0878 ! 


e 


c 


_ 


49°73t 


- '00262! 


-•08581! 


ee' 


c + m 


+ 


1455 t 


+ '0016 t 


- '2090 ! 


ee' 


c - m 


- 


1829 t 


-•0088 t 


+ ■0338 ! 


ee' 2 


+ 2m 


+ 


0984 




- '4034 


ee' 2 


c - 2m 


- 


1817 


- '0282 


4- '2009 


ee' 3 


c + 3m 


+ 


01 




-•85 


ee' 3 


e - 3m 


+ 


06 


+ •28 


+ '04 


e 2 


20 


- 


0618 


- '0007 


+ 0758 


eV 


20 + m 


+ 


0371 


4-40008 


+ '1794 


e%' 


20 - m 


- 


0460 


-•003 s 


4- -0027 


k 2 


2g . 


+ 2 


9897 


4- "0090 


— '2062 


kV 


2g + m 


- 


1092 


-•0057 


- 'S989 


kV 


2g - m 


+ 


0985 


+ '0409 

2Z 2 

a 2 


+ '2296 


1 





- 


00803 


+ "00028 


+ '00028 


e 


in 


- 


00005 


- '0003 3 


+ '00083 


e 





+ 


00386 


+ '00014 


+ '00029 


ee' 


c + m 


- 


0014 


- '0002 


+ '0007 


ee' 


c - m 


+ 


•0012 


4- '0004 


'OOOO 


k 2 


2g 


+ I 


9997 


+ '0060 


-■1479 


k%' 


2g + m 


_ 


•0646 


- '0038 


-•4377 


kV 


2g-m 


4 


•0762 


+ ■0273 
d (us — 2Z 2 \ 


+ '1608 




e dA a 2 ; 




1 





+ 


'00922 


-•00057 


-•00057 


e' 


m 


+ 


•00183 


4- '00043 


- '00177 


e' 2 


2111 


+ 


■0021 


+ 0013 


- -0042 



For other arguments, 

d (US - 2Z i \ us - 2Z 2 



de\ a 2 / a 2 

with sufficient approximation. 



x index of e in char., 



n d 

— -r-tllS - 22") 

a 2 dn> ' 



-1-32383! 

- 0240 ! 

- '022 

+ "5925 

+ -56.19 

— '264 

4- -6oi 



+ 



02538I 

01 5 1 j 

000 

0008 

0070 

0348 

001 

116 



+ 

-1- 
+ 



02538T 
1 140 ! 

354 

2220 

5i 6 3 
°53' 
99i 
75 6 



•0986 


+ -0008 


- 


•2228 


'100 


-■003 


- 


•440 


•131 


4- '012 


_ 


•■t77 


'9533 


-■0310 


+ 


•4638 


•018 


4- -018 


+ 


-"5°3 


•016 


-•145 

~v T~\- 2Z 
& A dn x ' 




•526 


•00800 


— '00064 


- 


•00064 


•0001 


+ '0007 


_ 


•0021 


•0034 


- -0030 


- 


•°°5S 



-r 9 888 






- -0208 


+ '3396 


- '012 






4- "012 


+ i"n6 


- '020 






- '096 


- -376 




d, 


'us 


- 2» 2X 

a 2 / 


62 2 


k 


lk\ 




a 2 



with sufficient approximation for terms 



k ^. 



I (US - 2??\ US - 2Z- 



lk\ a 2 



index of x i 
k in char./ 



for terms containing argument 2g. 



IP 










Motion of 


the 


Moon, 






19 








2 88. 


Value of —>u 2 £~ 
■ J a 


2 


Coefficient of £" ! ' 2i x chai 


xicteristic. 








Char. 


8 




1 = 2 






i= 1 




1 = 




•/ = - 1 


i = - 2 1 






I 


O 










+ '0043! 




+ '9879t 




- -0140! 


+ 'OOOlf 1 






e 


m 










- '0034! 




- -i86ot 




- '06l2"f 


+ -ooo6t 1 






e 


- m 




+ '00021" 






+ -oi82f 




+ '19611 




+ '0056! 


- 'OOOlf I 






e' 2 


2m 










- -0026! 




- 'io39t 




- -1817! 


+ •003 it 1 






e' 2 


- 2m 




+ '001 3! 






+ "°553t 




+ -i535t 




— 'oo66t 








e 


e 




+ - oooit 






+ -0046! 




+ "4979t 




- '2982f 


+ '0024"f I 






e 


- c 




+ 'ooiof 






+ ■1072! 




-i"4933t 




+ '0123! 








ee' 


e + m 










- -0039 




- "2342 




- '7475 


+ '0167 1 






ee' 


- c - m 




+ •0057 






+ '3 2 34 




+ -1403 




- '0090 








ee' 


c - m 




+ '0003 






+ •0186 




+ -303° 




+ -0499 


— -0019 1 






ee' 


— c + m 




— '0009 






-•0581 




- '3 106 




+ -0576 








ee' 2 


e + 2m 














— "ii 




-i'53 


+ '07 1 






ee' 2 


- e, - 2 m 




+ •03 






+ 75 




+ -13 












ee' 2 


c - 201 










+ •06 




+ '34 




+ "57 








ee 2 


- c + 2111 










- '20 




- '35 




+ -18 








e 2 


2C 










+ ■0033 




+ '2507 




— '2069 


+ •0259 1 






e 2 


- 20 




+ "0112 






-■0519 




+ '6234 




- '0002 








e 2 e' 


2C + 111 














- -19 




- '45 


+ •13 








e 2 e' 


- 20 - m 




+ •06 






-■is 




- -24 














e 2 e' 


20 - m 










+ '02 




+ '25 




— 'II 


- 'OI 1 








e 2 e' 


- 2c + m 




- 'OI 






+ ■07 




+ '37 












1 


k 2 


2g 














+ '0030 




- -1861 


+ ■0033 








k 2 


- 2g 




+ '0005 






+ 0896 




+ 1-9743 




- '0142 








1 


kV 


2g + rn 


















- '4i 


+ '02 1 








k 2 e' 


- 2g — m 










+ ■23 




+ -28 




+ -oi 










k 2 e' 


2g — m 










-•l6 




- '34 




- -06 




IB 




1 


k 2 e' 


— 2 g + m 














Value of ~ 2 ^X u0 "'C 


2 ) 


+ -24 


- 'OI 1 






1 


i 













- '00005 




- -00806 




+ '00073 








,/ 


m 










+ 'OOOI 




+ '0046 




+ '0022 








e 


- m 










- '0002 




- -0050 




— 'OO06 








e' 2 


2111 














+ '002 




+ 'OO5 




^■Ifilfl 




e' 2 


— 2m 










- -ooi 




- '004 


















Fo 


• other arguments 


multiply 


by 


power of k iu characteristic. 




Bi 






























Hi 

B i 
































S^^l 

































... «.,.. , -ivwyA « ■ vawva ' wwp~ _ - ■■ gp^ 



20 



Mr. Ernest W. Brown, Theory of the 



d 



Value of -, -Wm^ -2 ). Coefficients of C +u x characteristic. 



Char. 


9 


1 = 2 




i= 1 


4 = 


i 


= - 1 


-t= -a 


1 





- "oooi4f 


- 


•oi 4 54t 


- r3IS48t 


+ 


•05410-!- 


- '00040T 


e' 


m 


+ -O002 f 


+ 


•0104 t 


+ *355 2 t 


+ 


•2420 f 


- -0040 f 


e 


- m 


- -0014 t 


- 


•0686 t 


- -3784 t 


- 


•0.254 t 


+ '0006 t 


e' 2 


2m 








+ -116 


+ 


•820 


- '022 


e" 2 


- 2m 


- "002 


- 


T;4 


- -234 








e 





- '0002 


- 


•°i33 


- '6255 


+ 


•7981 


— '0122 


e 


— c 


- -0039 


- 


■2956 


+ 1*8414 


- 


•0450 


+ 'OQOI 


ee' 


+ m 


+ '0004 


+ 


•0164 


+ -5468 


+ 2 


•0102 


- '0842 


ee' 


— c - m 


- -0304 


_ 


•9260 


- '6502 


+ 


•0366 


- '0002 


ee' 


c — in 


- "0OI4 


- 


•0738 


- -8300 


+ 


•3428 


+ ■0058 


ee' 


- c + m 


+ '0030 


+ 


■0148 


+ I c 3722 


- 


•2232 




ee' 2 


c + 2m 








+ '190 


+ 4 


'344 


- -364 


ee' 2 


— c - 2m 


- -140 




•292 


- -616 


+ 


•004 




ee' 2 


- 2m 




- 


•272 


- 1 '064 


- 2 


'092 


+ '030 


ee' 2 


- c + 2m 


+ •008 


+ 


•784 


+ 1-650 


- 


•748 




e 2 


2C 


— "0002 


- 


■0116 


- -2846 


+ 


*545 2 


- '1028 


e 2 


- 2C 


-•0458 


+ 


•1216 


- 7044 


+ 


•0006 




e%' 


2C + m 




+ 


•016 


+ -458 


4-1 


■130 


-•540 


e 2 e' 


- 20- m 


-•268 


+ 


•348 


+ 740 








e 2 e' 


20 - m 


- -002 


- 


•070 


- -676 


+ 


•976 


- "022 


e 2 e' 


- 2c + m 


+ - oo8 


_ 


•270 


- 1 '306 


+ 


•006 




k 2 


2g 




_ 


•0002 


- '0076 


+ 


'333 2 


- '01 12 


iv" 


- 2g 


- '002 2 


_ 


■1502 


-1-9150 


+ 


•0456 


- -0002 


kV 


2g + m 








+ '004 


+• 


•726 


- -070 


kV 


- 2g- m 


- '014 


- 


■374 


- -484 


_ 


•020 


+ - 002 


k 2 e' 


2g-m 








- -008 


- 


•504 


+ '022 


kV 


- 2g + m 


+ '004 


+ 


■366 

e 

Value of -5 

ar 


+ -68o 


+ 


•194 


+ '002 


1 





+ -00003 


+ 


•00249 


- -01584 


+ 


•00612 




e 


m 




- 


•0021 


- -0090 


+ 


•0176 




e' 


- m 


+ "00003 


+ 


•0082 


+ '0025 


- 


■0028 




e" 


2 m 




- 


•005 


- -009 


+ 


•040 




e' 2 


- 2m 


+ -oo t 


+ 


■020 


+ -004 


- 


"012 





For other arguments, multiply by power of e in characteristic. 



Motion of the Moon. 



21 



289. Coefficient of ^ x charac- 
teristic in 

w^- 1 -Hi n d 







a 2 


a 2 


dnS" 


4 • 21 


Char. 







Coef. 




k 


g 


+ 


•998 


- 1 


•1646 


k 


-g 


— 1 


•0003 " 






k 3 


-g 


+ 1 


'99 


+ ! 


■1681 


ke 2 




- 


'5°3 






k 


g+2 


+ 


■0030 


- 


•0108 


k 


-g-2 


+ 


•0072 


- 


•0246 


k 3 


-g-2 


- 


•035 






ke 2 


-g-2 


+ 


■063 






k 


g-2 


- 


•0457 


+ 


•1213 


k 


-g + 2 


+ 


•°35S 


- 


•0857 


ke 2 


- CJ-+ 2 
! 


+ 


'027 






ke 


g + e 


+ 


'503 






ke 


-g-e 


+ 


•496 






ke 


g-e 


- i 


■487 






ke 


-g + c 


+ 


•480 






ke 


g + c - 2 


- 


•280 






ke 


-g-C+2 


+ 


•036 






ke 


g-e + 2 


+ 


•108 






ke 


- g + c - 2 


+ 


•088 






ke 


g-C-2 


+ 


•026 






ke 


-g+e+2 


+ 


•015 






ke' 


g + m 


- 


'1082 


+ 


■1439 


ke' 


g + m - 2 


- 


•1430 


+ 


•6456 


ke' 


g- m 


+ 


■1176 


- 


■1755 


ke' 


g - 111 - 2 


+ 


•0416 


- 


•1805 


ke' 


- g - m 


- 


•0835 


+ 


•1745 


ke' 


— g - m + 2 


+ 


•1066 


- 


■2825 


ke' 


-g + m 


+ 


•07 1 8 


- 


•1384 


ke' 


- g + 111 + 2 


- 


'°433 


+ 


'1084 


ke' 


-g+m- 2 


+ 


•0299 


- 


•1077 


ke 2 


- g- 2C - 2 


- 


•0766 






ke" 2 


g+2C-4 


+ 


•0198 







290. 



Coef of ?x 
char, in 



Char. 


±6 


Coef. 


I 





+ I 


•0046 


e 2 


O 


+ 1 


•286 


k 2 







•00 


1 


2 


+ 


•01794 


e 2 


2 


+ 


•481 


k 2 


2 




•00 


e 


e 


+ I 


'262 


e 


c+ 2 


+ 


■0427 


e 


e - 2 


+ 


■2504 


e 


c-4 


+ 


•0081 


e 2 


20 


+ I 


'244 


e 2 


20 - 2 


+ 


•2208 


e 2 


20 - 4 


+ 


■048 


k 2 


2g 




■0000 


k 2 


2g-2 


- 


"°39 




Coef of ^ in 




a 5 (r 2 


- 3* 2 ) 






r' 


a 2 









+ 


•994 




2 


+ 


•0132 




c 


+ 


•0823 




C - 2 


+ 


■0158 




20-2 


+ 


'00072 




2g 


+ 


'00613 




2g-2 


- 


•00036 



Coef of '( e x char, in 



u l /a\ a 



Char. 







1 


2 


+ -9915 


e 


c 


- '0697 


e 


- c + 2 


- '268 


e 


c + 2 


+ 1723 


e 2 


-20+2 


+ '004 


k 2 


- 2g+ 2 


+ 2'OI3 



» 2 0k = Coef. of £~ g in 

« 2 • ulT 1 ■ ?.i fa'v' 
_ ^ 

= - 1 '0062 t 

)\ = - 2 -1566 f 

h - + '012 1 7t 

h = + '99 2 ° t 



291. Coefficients of 

£" in 



2ȣ 


l -D(fz) 


ka 2 ( 1 + in) 


_{D+ 1 +n\)(u'C'- a) 


ka 


-( 1 + m) 





Coef. 


g 


+ '0O3 


-g 


+ 2 •0052"!' 


g+2 


- '001 


-g-2 


+ -006 


g-2 


+ •°5 2 9 


-g+2 


+ '0017 



With factor «a 2 k 
and argument = - g 



•170-r 



/„ + '00026 



+ I'OOOOT 



h 
h 
k 



;i^^^^^^^^^^^^^^j gppggi ppippjpg|gg pj| p p^ ^|jg^SPIgp 



22 



Mr. Ernest W. Brown, Theory of the 



292. Values of M^ j^M,,, j 2 M i} j ?J M t . 

To obtain the values, each coefficient is to be multiplied by its characteristic, 

M lf M 2 are unaltered, and M s changes sign when the angle changes sign. 

Coefficients followed by the mark f have characteristics of two orders higher included. 



Char. 


Angle. 




Mi 


M 2 


M 3 


iJh 


kX 


hx t 


1 





+ 


99276"!" 


- '0280! 





- 1-3240! 


+ -io8ot 





e 


r 


+ 1 


'0005 1 


- '0835! 


- -0107! 


-1-3478! 


+ '3 2 4St 


+ -0509! 


e' 2 


2 r 


+ 1 


2 59T 


- '201! 


- -040! 


- 1-701 


+ -852 


+ -302 


e' 8 


•7 


+ 1 


•65 






- 2'2I 


+ 2'0 


o'o 


e 


1 


- 


•49080! 


- '2862! 


- -3108! 


+ '5926 


+ 7523 


+ '8423 


ee 


i + i' 


- 


•3542-r 


- '421 


- '477 


+ -18 13 


+ i - iii7 


+ 1-3085 


ee' 


i-v 


- 


•6736! 


- '799 


- -889 


+ I-I544 


+ 2-5564 


+ 2 9I24 


ee' 2 


1 + 21: 


- 


•370 


- 73 


- -8-> 


+ '065 


+ 2-151 


+ 2-509 


ee' 2 


1-2I 1 


- 


•978 


- I'lO 


- 1 '34 


+ 1*904 


+ 3'595 


+ 4' 6 43 


e 2 


21 


- 


•0618 


- -2069 


- -2065 


+ -0986 


+ '5453 


+ '544i 


eV 


2I + V 


._ 


•0247 


- -24 


— -24 


- -ooi 


+ -586 


+ -582 


eV 


2l-v 


- 


•1078 


- 73 


- 73 


+ -230 


+ 2 -6 1 4 


+ 2-604 


1 


2D 


- 


'00701! 


+ "9870! 


+ -9868! 


+ -0254! 


-i - 3 I 44t 


- 1-3136! 


e' 


2D + 1' 


- 


•00291! 


-1*17351' 


-1-1739! 


+ -0103! 


+ 1-6697! 


+ 1-6709! 


e' 


2D - 1: 


- 


•03693! 


+ 3-1571! 


+ 3-1561! 


+ -1394-t 


- 4'3 2 4it 


-4-3169! 


e- 


2D + 2l' 


_ 


'0042! 


+ '0826! 


+ -0818! 


+ -017 


- "239 


- " 2 39 


e'- 


2D - 2 1' 


_ 


•1265! 


+ 7-1584! 


+ 7'i534t 


+ -500 


- 9-928 


- 9-892 


e 


l-2l) 


- 


•08582! 


- 1-4897! 


+ i'4944'f 


+ -2220 


+ 1-8274 


- 1-8518 


ee' 


I-2D + 1' 


- 


•2948! 


-4-321 


+ 4 - 349 


+ 7'383 


+ 4797 


- 4-941 


ee' 


1-zD-l' 


- 


•0520! 


+ n88 


- 1-178 


+ -275I 


- 'S 01 


+ '44° 


ee' 2 


I- 2D + 2/' 


_ 


■7197 


- '911 


+ 9-21 


+ 1785 


+ 9-01 


-9-67 


ee' 2 


I-2D-2I' 


- 


•0716 


- '03 


+ -05 


~ '425 


+ '24 


- '31 


e 


I + 2J) 


- 


■0026 


+ '4972 


+ '4972 


- "0008 


- -6248 


- -6246 


ee' 


I + 2D + I' 


- 


'00 10 


- 73 2 


- 732 


- -0078 


+ 1-1719 


+ 1-1717 


ee 


I + 2D-I' 


_ 


'0114 


+ 1795 


+ 1-795 


+ '0340 


- 2-7040 


- 2-7038 


e 2 


2I - 2D 


+ 


■0758! 


+ -6487! 


- '5969t 


- '2228 


- -8064 


+ '6010 


eV 


2I-2D + V 


+ ■ 


'2552 


+ 1-73 


- i'53 


- -663 


- r8i 


+ "93 


eV 


2I-2D-- V 


-4- 


•0785 


- -18 


+ '32 


- '400 


- '93 


+ -27 


k 2 


2F 


+ 2 


•9901 


- '2001 


- -1717 


-2-9549 


+ -3785 


+ '2873 


kV 


2F + r 


+ 2 


•8805 


^ ' 2 5 


- -19 


-2-937 


+ '5°9 


+ -275 


kV 


2 ¥ - r 


" r 3 


•0882 


- '37 


- '27 


-2-939 


+ -645 


+ "349 


k 2 


2 F - 2D 


- 


■2062! 


+ 1-9756! 


- 1-9690! 


+ -4637 


- 1-9243 


+ 1-9019 


kV 


2F-2D + Z' 


- 


■8051 


+ 6'22 


-6-i8 


+ 1-967 


-6-282 


+ 6-164 


kV 


2F-2D-Z' 


- 


•1653 


-2-31 


+ 2-31 


- -319 


+ 2-583 


- 2-605 



Motion of the Moon. 







Values of 


M u 


JiMi, j\M,; 


j 9 Mi- 


-continued. 




Char. 


Angle. 




h.M, 




:kM, 




hM-i 


JJf, 


J s m, 


I 


o 


+ 


'00922 


+ 


'01223 







- '02415 


+ '00146 


e 


/' 


+ 


•oi 105 


+ 


'02710 


- 


■00403 






e" 2 


2l' 


4- 


' OT S4 


+ 


■0413 


+ 


•0032 






i 


2D 


- 


•00057 


- 


•01583 


-- 


'01583 


+ '0008 


- '0081 


e 


2D 4-/' 


- 


'000 [4 


+ 


•0685 


+ 


■0685 






e 


2D-I' 


- 


•00235 


- 


•04494 


- 


'O4490 






e" 1 


2D + 2I' 


+ 


"oo 10 


+ 


'0004 


+ 


■0004 






e 


2D - 2/' 
/ 


- 


•0067 


- 


•0918 


_ 


•0916 


+ '01 16 





JsMs 
o 



Char. 


Angle. 




M 1 




M., 




M s 


^6 


M 7 




J/ 8 


M s 


a l 


D 


+ 


'1132 


+ 


'3S°i 


+ 


■3681 


+ '972 


+ I'OOO 


- 


'007 


- '007 


a t e' 


D 4- 1' 














+ '380 


+ '435 


- 


■194 


- '195 


a l e ' 


D - /' 














+ 2'S°5 


+ 2-639 


- 


•029 


- '029 


a^e 


I - D 


+ 


•00396 


- 


•3!363 


+ 


'26736 


— 1 '464 


+ I'002 


+ 


•0139 


- -0088 


o.f.e' 


1-D + V 














-3 '53 


+ 2-35 


+ 


•056 


- '°33 


a^k 2 


3D - 2F 


+ 


•0951 


+ 


'977 


+ 


•983 


~ r 57 


- i'69 


+ 


•980 


+ -980 


a,kV 


3D-2F + // 














+ '62 


+ '62 


- 


3 '48 


-3'48 




D~ 2 Z 


- 


•1387 


+ 


•179 


- 


•131 












o-je 8 


3/ - 2D 


+ 


•02080 


- 


•01282 


4- 


•06138 












ajC 3 


4D-3Z 


- 


'00226 


■- 


'02460 


~ 


■02394 












c^ek 2 


I-2Y 




3'497 


+ 


•2952 


- 


•6828 












a,ek 2 


4D-/-2F 


+ 


■01108 


+ 


•01113 


+ 


'01 169 












Ojk 4 


4 D - 4 F 




•00 


- 


■0169 


- 


•0165 













Jhar. 


Angle. 




ih 


iA 


k 


2 !0j - w 3 — h" 


+ 1 


•004 


- i'i65 


ke' 


2W 1 - w 3 - h" + /,' 


+ 


•896 


- 1 '021 


ke' 


2 w 1 - w a - h" - V 


+ t 


•Il8 


- r 34° 


k 


w a - h" 


- 1 


'0040! 


+ ri68tt 


ke' 


w 3 - h" + 1' 


- 1 


•08 7 1 


+ i'343t 


ke' 


w s - h" - I' 


- 


'93 2 t 


+ i'03ot 


k 


2T -w s -h" 


- 


'°4S7 


+ '1213 


ke' 


2T — ic H — h" + I' 


- 


•1887 


+ '7669 


ke' 


2T - w s - //" - V 


- 


•0041 


- "°59 2 


k 


2 D + m; 3 - /i" 


+ 


•°3SS 


- '0857 


ke 


2D + t« 8 - A" + I' 


+ 


•1421 


- -3682 


ke' 


2D + ?» 3 - h" - r 


- 


•0078 


+ '0227 



Char. Angle, 

ek w x + w - ir B - It',' 

ek w^ — w„ + w z — li," 

k 3 - 2>i\ + 3?o s - h" 

e 3 k w 1 4- 3« 2 - w B - h" - 2I 



- 1 -49 
+ '49 

- '989 

- '0154 



... — ■ ■•■••ff^^.'.vAv.^vrv^^r»^.^^^9-'^-r,-'.i.'^T' '■'■'• '^ 



24 Mr. Ernest W. Brown, Theory of the 



CHAPTER XL 

THE DIRECT ACTION OF THE PLANETS. 

Section (i). The Disturbing Function. 

293. Axes and Notation. — The value of the disturbing function and the equations 
of variations are the same whatever axes be chosen. For the expansion of the former 
I take for plane of xy the ecliptic of 1850-0, and for axis of x a line parallel to that 
joining the Earth and Sun, that is, a line parallel to the Earth's true radius vector on 
the assumption that the Sun moves in an elliptic orbit. With these directions, x, y, z, r 
will now represent the coordinates and distance of the Moon, £, n, £, A those of the 
planet, the Earth being the origin. 

For the elliptic coordinates of the Earth and planet with the Sun in the focus, I 
take r', r" (the distances) ; V, V" (the true longitudes) ; T', T" (the mean longitudes) ; 
xs', vs" (the longitudes of the perihelia) ; o, h" (the longitudes of the nodes) ; o, y" (the 
sines of half the inclinations ; e', e" (the eccentricities) ; 2a! , 2a!' (the major axes). All 
these longitudes are measured in the usual way, that is, from a fixed line in the plane 
of xy to the node, and then along the plane of the orbit. 

The perturbations of the planet's orbit, like those of the Earth, axe neglected in this 
chapter. In order that the motion of the Earth round the centre of mass of the Earth 

and Moon may be taken into account, the terms depending on a/a' in the disturbing 
function must be multiplied by the ratio of the difference of the masses of the Earth 

and Moon to -their sum (Chap. I., §4); we must therefore use a x a/a instead of a/a' 

(Chap. IX., § 193), or, with sufficient accuracy, a t instead of a. 

The masses of a planet and of the Sun are denoted by m", m', and the mean 

longitudes of the planets, measured like the other longitudes, as follows*: Mercury, Q ; 

Venus, V ; Earth, T ; Mars, M ; Jupiter, J ; Saturn, S ; the other planets and the 

asteroids will not be considered in this chapter. 

294. The Disturbing Function R and its Transformation. — -From Chap. I., 

Sect, (i), we obtain, on changing from the Sun to a planet as the disturbing body, 

R ....... x _x^ + yr l + zt A 2^p + v 2 + p . . . (1), 

to" {{£ - xf + ( v - y) 2 + (£ - zf}* A :i 

* Mean longitudes are denoted by roman capitals. 

t A.P.E., sect, it, where the full details of the transformation will be found. 



Motion of the Moon. 
If we denote by d/dQ the operator, 



3 3 3 3 

l)Q of o-q o£ 



the expansion of R may be put into the form 

R 



I C A _ I 0" 

2T 3CP~ 3! 3a 3 ' 



or 



, since 1/ A is a solution of Laplace's equation, so that 



0" I 

id a 



into the form 



3f 2 St; 2 / A 



— ).?-- + terms of hieher degree 



\c'£* oj-v \c£- 07]-/ ti^crj \ ct crj/ oQ 

Now, in polar coordinates, we have 

£ = _,-' + (!_ y"2) r " cos ( F" - V) + y" 2 r" cos ( V" + V - 2I1") , 
7] = ( 1 - y" 2 )v" sin ( V" - V) - y" V sin ( F" + V - 2/1") , 

£ = 2y"(i - y" 2 )V" sin ( V" - h") 

Hence 3/3£ = - 3/8/, and if/ be any function of if + >? 2 and £, 

5/' ,3/ ,3/ , 3/ ,3/-' 

„—} = - r X- ~ k-- + '1X-- = ~ r X > 

3 V 07] ' 07] Ct 07] 



00 : 



(3), 



JL ( 4 ), 

A 



(5), 



r 3 V"- 



?■-— + < 

d?; 0?; 



3\5/' ,3 2 / c f Xof c/'\ of ,c 2 f 3/ 
o^/ct] C7f CTj\ ct] C£/ eg c^ cr 



?l_i!_. ^'-JL ^ iS^. 8 2 /_5 / / i 3/ 
>2 ~ S,.'2 ' TXXi ~~~ Xa> -XXX'i + „.' 2,.' 



Hence 

3f 2 3r' 2 ' ct; 2 '; /2 3 V' 2 ' r or ' o£otj cr'\r' oV 

Further, since (cr/dp + <f/d>f)f is a function of £ 2 + >? 2 , £, only, 



crj\o£ 2 erf J r 3 l"\3r' 2 r' 2 V" 2 r' or' 



(6). 



(7), 



and since we may differentiate each of the equations (6) with respect to £, that is, to 
- /•', we can find all the derivatives of the third order with respect to £, n in terms of 
derivatives with respect to r', V ; and so on. 

Again, by expressing A in polar coordinates, I have constructed the formula * 



° I _ _ "/' e< v '- ]i ") t 3 1 3 , _ » 2 \_^ I l 

U, A" " {T-^y^y 1 ~~i-'"~ \ 3F' + S/ 2 3/7 i l " y V' 2 J ^ 



(8), 



which may be combined with the previous equations, since the left member of (8) 
shows that the right-hand member is a function of if 2 + i? 2 , £, and that its imaginary 
part is zero. 



* A.P.E., p. 19. 



Jaoyal Astron. Hoc. Vol. LIX. 



^^w^^^^^W^^^^PPPP^PPPPBPIll! 






26 Mr. Ernest TV. Brown, Theory of the 

Finally 

,3,333 , x 

cr oa cv el 

and therefore if i/A be expanded in terms of the elliptic elements of the Earth and the 
planet, the functions needed are all expressed as derivatives of i/A with respect to the 
elements present in the development. It is true that these derivatives are to be 
multiplied by 1/r' 2 and that the lunar coordinates have to be transformed so as to be 
referred to the true instead of the mean place of the Sun ; but the work needed to 
perform these two operations is very small, especially when compared with the labour 
of making developments of several different planetary functions, such as 



A3 a 5 ' A 5 ' /\ s ' 



etc. 



295. The transformed Disturbing Function. — I omit the algebraical details 
necessary to follow out this method, so as to present the results in a form convenient 
for numerical application. The result is a form for the disturbing function expressed 
as a sum of products : the first factor in each product is a function of the Moon's 
coordinates, u, s, z, multiplied by a certain function of r' , V ; the second factor consists 
of derivatives of i/A with respect to a', T, h", y //2 . 

The notation for the first factors has already been given in § 285 of the last 
chapter. If an angle 9 is present in the first factor, then the corresponding term is 
M t cos, 6 + iMi sin 6, where such constant factors have been taken out that the M { may 
be numerical quantities. 

296. For the second factors, let a term with argument <p in i/A be 



— = P COS 1 



If i, i! be the multiples of T, 2h" present in <p, the operators 9/3 V, djdh" give rise to 
the factors i, i', and the cosine is changed to a minus sine. 

Also P is of degree - 1 in length, and may be expanded in the form f{a)/a', where 
a = a'la", or in the form f(a)ja", where a = a"/a', according as the orbit of the 
disturbing planet is outside or inside that of the Earth. If, then, we put /= ad/da 
for a derivative with respect to log a only in so far as a occurs explicitly in P after 
P has been expressed in one or other of these two forms, we have : 



(11). 



When we combine the two factors, we obtain the product of two cosines or two 
sines multiplied by a constant factor ; this product is expressed as the cosine of the sum 
or difference of the two angles 0, <f>. The notation for the planet coefficients is as 
follows : — 



For outer planets, 


r dP 


d 

= a-r" = I , 

da ' 




a 
a = — , 


P= 


'k&-> 


For inner planets, 


d 


d 

= — a.— - 1 = 
da 


-I- 1, 


a" 

a _ 

a ' 


P-- 


=>o 



Motion of the Moon. 
Let J + , »/_ denote the operators 



27 



oy - \ y "/ 



and let *', * 7 denote the multiples of T, 2W present in <f>. 
For outer planets put 

P^iP-i^P, P^l-P^-lP + PP, P 3 = -i(l- i)P, P 4 ±P 5 = -(l±i)J±P . 
P a ±P 7 = (I-2 + i)P 1 , P B ±P g *=(l+ 3 i-6)P 1 + 4(i±i)(2±i)(I+t)P, 

P W ±P U = J±( - A + 2lP±2iP) , P l2 ± P ]3 = - / ± {Pj + ( ± 2i - 2)IP+{2t l + 2«')P} 

all the upper or all the lower signs in any equation being taken together. 

The product of the pairs of factors gives for any term of argument 6±(p in R, 



(12), 
(ISA 



,, Ilk m ; ,1 

R = — n -&- ■ aa 
4m 



M.P, + M 2 P 2 + M S P S - ^7 M^ ± P f) ) 



+ \a x (M 6 P 6 + M 7 P 7 + M S P S + M 9 P 9 ) - y a \,.,. {M, (P w ± P n ) + M 12 (P 12 ± P 13 )} 

2(l-y')= 



cos(0±<fi) . (14), 



all the upper or all the lower signs being taken according as it is convenient to use 
Q + (p or 9 -cf>. 

For inner planets, we replace aa by a' in this formula, and /by —I—i 
in (12), (13). 

It is to be noticed that Pfi" for outer planets and Pft' for inner planets are 
numerical coefficients when the value of a has been substituted. 

297. Method of computation of the coefficients in R. — The terms in R with 

suffixes i, 2, 3 give rise to nearly all the sensible perturbations of the Moon's orbit. 

In the great majority of cases M % P^M^P Z is small compared with either of the two 

terms, and this is due to the approximate numerical equality of M 2 and M z and of P t 

and P s . It is therefore better (and the computations were so made) to use these two 

terms in the form 

i(AT 2 + M :i )(P 2 + P s ) + i(M 2 ± M t )(P 2 - P s ) ; 

and then again one of these two expressions was generally small compared with the 
other. Another advantage of this form arose from the fact that it was sufficient to 
have M 2 , M B to four significant figures and JP 2 , P s to six, instead of both to six. And, 
moreover, the near equality of P 2 , P s can be foretold by the theory, when it exists. 
A similar circumstance holds with the pairs with suffixes 4, 5; 6, 7; 8, 9; 10, 11 ; 
12, 13. 

There is not a large number of terms depending on the terms in R with suffixes 
4. 5, fewer still with 6, 7, 8, 9, and none sensible with 10, 11, 12, 13. It did not 
seem necessary to carry the computations to the next term of R, depending on a x 2 ; this 
coefficient gives a factor 6 x icr 6 compared with the first terms of R, and it gives rise 
to no arguments which are not present in the first terms. 



28 Mr. Ernest W. Brown, Theory of the 

Section (ii). The Computation of the Coefficients P { * 

298. Leverriers Expansion of 1/ a. t— Leverrier's literal expansion in powers of 
the eccentricities and mutual inclinations, with coefficients depending, on a and arguments 
on T, xs', m", h", was used. Here the Earth is supposed to move in the plane of 
reference, and the notation is slightly different. I have, therefore, put in the 
development — 

For Leverrier's symbols, t, t', /', X, <■>, a, «■, V 

the symbols h", //", 'J', T",i ~", «", a, y 

for inner planets, and have then interchanged the accents of a, rs, e, T for outer 

planets. 

Leverrier's development contains functions of a through certain coefficients (3%, 

which are defined as follows. Put 

A", = 1 + a- - 2 a COS (T - T") , j 



. = W2" OC j8--'cosi(T-r') ) <C= ^:£#' 



\ ■ ■ ■ (is)- 



i_i 



A," 2 ,: -co ' ■' p\ da 

Instead of ft, he uses the letters A, B, C, D, ff, //,§ according as s = J, f , f , £, f, V; 
I adopted the same notation, as well as the following : 

H«> = t^/J"- 3 ' + 9D"- 1 '' + 9 Z>''+ l! + L> ii+3 <) , L"» = |(C ! '- a + a" 1 ) , 

to which the suffix p may be attached according to the previous definition. But 
I dropped the brackets round the indexes to the letters, since powers of these 
functions do not arise ; and the indexes themselves were also dropped whenever they 
were all the same, equal to i, in a given equation. 

299. Formula for computing the coefficients.— Leverrier gives the numerical 
values in most cases up to i=io, but they are needed much further in many cases 
of the lunar problem. Hence, all of them were newly computed by the formulas 
which are fully set forth in Section III. of my Adams Prize Essay. These formulae 
are constructed for several purposes : first, for finding isolated coefficients for special 
values of i ; second, for making tables of coefficients for many consecutive values 
of i ; third, for the avoidance of those small coefficients which appear as the difference of 
two large numbers. The third point is a difficulty which arises chiefly in computing 
(P-i*)^, the two parts of which are large compared with their difference, especially 
for large values of i ; but the formulas for these completely surmounted the difficulty. 

* A.P.E., sent. iii. 

t Aim. Uhs. Paris (Mem.), vol. i., where the expansion is given so as to include terms of the seventh 
order with respect to e', e", y". Boqukt {lb., vol. xix) has computed the terms of the eighth order. 
I Denoted P in A.P.E., by inadvertence the same letter as for the general coefficient. 
§ Leverrier does not need the last two. 



Motion of the Moon. 



29 



It arises in a less troublesome form in P 2 , P 3 , but there the number of places of 
decimals computed was always sufficient for the degree of accuracy required. 

300. Numerical values of A' p , B],, . . . — These were computed to six significant 

figures in tables * as follows : — 

o 

Venus : A I Bl to i = 43 ; A), Ii), to p = 4, 1 = 30 ; C), to p = 2, i = 30 : 

D M L\, 3"„ to i = 30 ; { (/" + i) 9 - * 2 ML t0 i J = 3> *'= 3° ; 

{(/+ i) 2 - (i+ i) 2 }-#j, top = 2, from * = -30 to i= 30; {(/+ i) 2 -i' 2 }C"', to ji? = i, 1 = 30. 
Jupiter to i = 6 for A), to p = 3, B\ to _p = 3, 0), to p = 1, Z>j. 
Mars: A\, BI 10 i = 30 ; and to « = 6 for A\, A\, B[, Bl,, C;. 
Mercury to i=-- 8 for .4],, i?]„ to /; = 4, Cj to jo = 2, Z>j. 

All other coefficients required with these planets, and those for Saturn, were 
separately computed as the needs for them arose. 



Section (iii). The Sieve.* 

301. The larger number of the terms in Ii which give rise to sensible coefficients 
in the coordinates have periods which are comparable with the month or the year. 
For such terms the obvious plan was to take the successive values of 6 (the argument 
arising from the lunar factors) according to the magnitudes of the coefficients which 
accompanied this factor. The arguments o, I, 2D - 1, 2 1), . . . were successively combined 
with all the possible arguments <f> until the terms became insensible and it was 
unnecessary to proceed further. In each case the argument <j> was divided into the 
series i(T — V), f(T- Twits', and so on, the magnitude of each series mainly depending 
on the power of e' , e", j" which accompanied it ; and again each of these series was 
computed with a sufficient number of values of i with each Moon argument. A little 
practice quickly enabled one to choose out the largest coefficient in each set, and a rough 
calculation was sufficient to show whether the term would be sensible. This rough 
calculation had to be made for both hv x , $w 2 . and sometimes Sio 3 , for, with terms of 
short period, the secondary inequalities (§ 282) are frequently sensible when the primary 
might be neglected. 

This method could not fail as long as the periods were short. These periods only 
arise in the equations of variations (§ 269) through the divisors s, s 2 . But if any 
period was long, then s, s 2 would be comparatively small, and a large coefficient might 
result. Such cases as occurred during the progress of the calculations were naturally 
dealt with as they arose ; the only matter which called for attention was the necessity 
for a larger number of significant figures in the coefficients M it P { . If, in the general 
method, the numbers were not sufficiently accurate, the special coefficient was separately 
computed again with more places of decimals. 

But the lunar terms contain multiples of four different arguments, w 1 — T, iv 2 , iv 3 , 
T-tir', that is, combinations of four different periods, aud the planet terms two periods, 
1 '. T. Hence we may have combinations of five different periods, and there will be 
* A.P.E., sect, v., where the numerical values will be found, * A.P.E., sect. iv. 



,v*.f7.K'*.'*rfC>S.::- f A.9<S'7s!>rwi'.'9WfXy- 



30 Mr. Ernest W. Brown, Theory of the 

many long periods arising therefrom. It was therefore necessary to sift out those 
which would give sensible coefficients. 

302. The Sieve. — The method was essentially the same as that for the short-period 
terms, but, as there were thousands of possible combinations, some plan had to be 
devised to find an upper limit to each coefficient so rapidly that every coefficient might 
be examined within a reasonable time. 

The limitations were as follows. Only primary terms were examined. It 
was shown* that the secondary would not be greater than the primary unless 
i x = o (§ 279), s>6o" or i x =i, s>iooo" ; the very few of the former were separately 
examined, and the latter had been treated in the short-period terms. Periods 
greater than 3500 years (s<i") or coefficients <o ;/ 'Oi were to be excluded ; neither 
could sensibly affect the motion of the Moon within historic times. But one or two 
longer periods with coefficients greater than o"'oi which appeared in the course of 
the work were retained. 

The possible long periods were then constructed by finding all up to the largest 
values of i, j in i(T - T") +jT, ±w x + i(T - T") +jT, ... It was soon seen that only a 
dozen or so in each set need be retained, and at the most three multiples of w x . Then a 
table was formed for the multiples of u\, w 3 , giving the periods, the lowest orders with re- 
ference to e, k which would accompany each multiple, and the multiple of T which would 
occur in the lunar argument with this lowest order. Thus, for a given multiple of w % , 
iv s , the various long-period combinations with the former sets could be seen at a glance. 

303. Next, very simple formulas were constructed for the primary coefficient, 
depending only on the power of e, k present in the lunar factor, the multiple of T-T" 
and the coefficient present in the expansion of i/a s or of i/A B . These formulas arose 
from transformations of the disturbing function somewhat similar to those of § 294, 
but depending on derivatives with respect to T only. The values of the coefficients 
in the expansions of A" 8 , A" 5 were obtained from Newcomb's tablet of these 
coefficients in the case of Venus; partly from his incomplete table, + and partly 
by extrapolation and by approximate computations, in the case of Mars ; for the other 
planets, which presented little difficulty, a table for A 3 was roughly computed, and 
simple formulas depending also on the order of the eccentricities and inclination were 
constructed. 

The various coefficients were examined according as they arose from terms whose 
characteristics were of orders o, 1,2,... With each order was associated a maximum 
value of s which could give sensible coefficients ; after the first three or four orders the 
work went very rapidly, as these maximum values of s became small, and the great 
majority of the terms could be excluded without computation. 

About 100 long-period terms were retained out of several thousand examined, and 
their coefficients were accurately computed. In no case did these coefficients exceed the 
preliminary estimates found by means of the sieve. No new terms of any great 
* A.P.E., p. 38. t Wash, Astr. Papers, vol. v., pp. 248-257. 1 L.c, pp. 258-261. 



Motion of the Moon. 31 

importance were found, and the corrections to Radau's values* of those previously 
computed were small from the observational standpoint, as far as the long-period 
primaries were concerned. 



Section (iv). Numerical Values of the Elements. 

304. Most of the observed quantities required are known with more than sufficient 
accuracy. The most doubtful is the mass of Mercury, which may be in error by as 
much as 50 per cent. ; but the largest coefficient with the adopted value is less than 
c/'-oS, and the term has a period of 39 years. The mass of Venus may be in error by 
1 or 2 per cent., giving a maximum possible error in the largest coefficient (period 
273 years) of o //- 3 from this cause. 

The values of the elements used are shown in the following tables : — 



Arg. 


i 

Daily motions 


Epoch 


Longitudes at Epoch. 


of arguments. : 1850-0. 


Perigee. 

75° 07' 19" 
129° 27' 34" 
100° 21' 40" 

333° 17' 55" 
11 54 27 
90° 06' 40" 


]S T cde. 


w 1 
iv 2 
w 3 

! Q 

V 
T 
M 
J 

S 


47434-89I 

400-923 

- I90-772 

I4732-420 

5767-670 

3548-I93 \ 
I886-5I8 

299T29 

120-455 


Mercurv 


46° 33' 12" 

75° !9' 47" 

48° 24' 01" 
98° 55' 58" 

112° 20' 51" 


Earth 

Saturn 





Moon e = ... 

„ e= ... 

Earth 


Eccentricity. 


Inclination. 


Sine halfinclin. 


1 «" 


■m' 

m" ' 


' I0 955 

•054906 

•016772 

•205604 

•0068446 

■093261 

•048254 

•056061 


7° 00' 07" 

3° 23' 35""3 
i° 51' 02" 
i° 18' 42" 

2° 29' 39" 

i 


k = '044780 

•y = '044887 

'061066 
■0296063 
'016149 
- oi 1466 
'022 


loga! = 3"39 88 

¥•5878216 

i'8593374 
'1828960 
■7162374 
'9794957 


6000000 

408000 

309350 

1047 '35 
3501-6 


Mercury 

Venus 

Mars 


Jupiter 



L.e. (§ 279), p. 113. 



?•'•'"•-• --^r>"w^^7^.-^rj?;-^^ 



32 Mr. Ernest W. Brown, Theory of the 

Section (v). The Final Results. 

305. The detailed results arising from each term of the disturbing function are 
fully set forth in A.P.E., Section vi., and they will not be reprinted. Many of the 
resulting terms in the Moon's coordinates, especially those arising from the short-period 
terms in R, have the same arguments, and must be combined. The final results only 
will be given here. 

There are two methods of expressing the perturbations. The first is to add them 
to the true longitude, latitude, and parallax of the Moon ; the second is to leave them 
as additions to the elements w v iv z , w z , a, e, y which would be tabulated with these 
additions. This latter method is only of special advantage for tabular purposes when 
the variations of a, e, 7 may be neglected, and this happens only with terms of long 
period in which u\ is present. If %v 1 is absent from the primary, and the period of the 
term is not very long compared with the periods of the Moon's node or perigee, the 
variation of e produces an effect of the same order as the variation of iv 2 , and that of 7 as 
iv n ; in fact, the statement in § 282 has to be remembered. In these cases the variations 
of a (or n) are insensible, and it may be convenient to retain the variations of tv x (or of 
iv v w 2 ) as elemental inequalities, adding the parts due to the other elements to the 
coordinates. No periodic variations of e have been retained as elemental inequalities. 

For certain other cases in which w 3 is present, with §e, 8a insensible, it is best to 
retain Sw lt Sw. it $w 3 as elemental terms, and account for those arising from Sy by 
multiplying the final value of the latitude by the variable factor 1 + Sy/y, and the 
terms in longitude containing the argument 2F by 1 + 2^7/7. But in setting forth the 
results such terms are left as perturbations of 7. Hence the terms are placed in two 
classes, those added to the coordinates, and, in addition, those added to the elements. 

The original limit set was o //# oi, but all short-period terms and most of the long- 
period terms have been computed to o" - ooi, and they are so retained here. A star 
replaces the last figure in the cases where the computations were only made to o"-oi. 

306. The tables are arranged according to the lunar arguments so that 6 remains 
the same until a new value is set down, and then according to the multiple j of 
T (or j" of T"), which again remains the same until a new value is set down, and 
finally, according to multiples of T-T". The coefficients are set down in units of 
o"'ooi, the angle a being so chosen (<36o°) that they are all positive. 

The value of the angle a is also not generally repeated when it is the same for a 
long series of terms. 







Motion of the Moon. 






H 






307. S V = + 0" -oo 1 sin { 6 +JT + i{T - V) + a } , Venus. 










= 


B=2J) 


6 = 


2D 






3 


i a 


3 i a 


J i 


a 


c IH^hI 




o 


I o'o 480 


0-15 I 


-1 - s 


84 


7 ifl^B 






2 200 


- 14 2 


- 4 


78 


7 IIH 






3 92 


- 13 2 


- 3 




4 Ufls 






4 60 


- 12 2 


- 2 




4 iH^H 






S 38 


-" 3 


- 1 




3 MHI 






6 25 


-10 5 


1 




1 I^^^^H 






7 l 7 


- 9 6 


-2-6 


162 


6 I1H 






8 12 


- 8 8 


i5 


'S 1 


4 ll^K 






9 § 


- 7 8 


18 


IS 1 


10 PHH 






10 6 


-6 11 












11 4 


- s 11 


6» 


=/ 








12 1 


- 4 10 


0-8 


i8o - o 


2 i[^^^Bl 






21 3 


- 3 i go 36 


- 7 




4 ifl^E 




. l 


- 3 92-2 1 


- 2 26 


- 6 




il^E 






- 2 4 


- 1 15 


- 5 




6 fl^E 






- 1 8 


1 15 


- 4 




9 liH 






1 47 


2 8 


- 3 




l6 B^^^^H 






2 272'2 76 


3 4 


- 2 




29 l^^H 






3 2I 


4 4 


- 1 




68 i:flH 






4 12 


5 4 


1 


O'O 


9i ;:'^H1 






5 7 


6 3 


2 




64 ili^^Ba 






6 6 


7 3 


3 


i8co 


127 it^^^Hi 






7 4 


8 3 


4 




7 l^HI 






8 1 


. 9 2 


5 




1 ll^^^Hl 




2 


- 18 209 50 


10 1 


22 




2 {f^^^HI 






- 4 27 i 


18 3 


1 1 


92 


8 - l^BI 






- 3 2 


1 20 273 3 


2 


272 


13 IBBI 






- 2 2 


-1 -15 78 1 


3 




6 IIBI 






- 1 3 


- 14 1 


4 




8 l^B 






1 6 


-■ 13 2 


s 


92 


4 I^BB 






2 8 


- 12 2 


6 




2 I^^HI 






3 37 


- 11 3 


2 3 


272 


6 l^HI 






4 207 8 


- 10 4 


-1 - 5 


268 


1 ^^Hl^ 






5 3 


- 9 4 


- 4 




Hi HB 






6 4 


- -8 4 


- 3 




i:I^P 






7 1 


- 7 5 


- 2 




13 I^BI 




3 


5 112 7 


- 6 s 


- 1 


88 


8 H^Hl 






Royal Astron. Soc, \ 


OL. -Lil-A. 


5 



















34 



Mr, Ernest W. Brown, Theory of the 



SV= + o" -oo i G sin { 9 +/T + i(T - V) + «° } , Venus. 





i 


a 


C 


I 


3 


280 


7 


2 


-18 


209 


3 




2 


27 


1 




3 




6 




4 


207 


1 




6 


198 


16 


2 


- 4 


333 


1 




- 3 


153 


6 




- 2 




1 




e= 


= 2 D-Z 




O 


-*3 


iSo'o 


1 




— 12 




2 




- II 




2 




- IO 




3 




- 9 




6 




- 8 




8 




- 7 




13 




- 6 




22 




- 5 




39 




- 4 




87 




- 3 




716 




- 2 


O'O 


I S 2 




— I 




74 




I 




13 




2 




10 




3 




7 




4 




S 




5 




3 




6 




2 




18 




11 




19 




2 


I 


- 1 


310 


2 


I 


- 10 


269 


1 




- 9 




2 




- 8 




3 




- 7 




S 





(9 


= 2D-Z 




i 


i 


a 


G 


— 1 


- 6 


269 


8 




- 5 




2 5 




- 4 


% 


33 




- 3 




10 




- 2 




S 




- 1 




3 




17 


253 


3 


- 2 


- 7 


340 


3 




- 6 


162-5 


83 




- S 


165 


4 




15 


151 


25 




e 


= zT> + l 







- 9 





1 




- 8 




2 




- 7 




2 




- 6 




2 




- 5 




3 




- 4 




1 




- 3 




4 




- 2 




4 




_ ! 




3 




I 


B=2l 


1 





- 3 


180 


1 




— 2 




2 




- 1 




S 




1 





s 




2 




2 




3 


180 


9 




d=2l-2D 







1 


i8co 


4 




2 




11 




3 




76 




4 





3 



:2Z-2D 



i 


l a 


c 


1 


4 92 


r 


2 


6 17-5 
= sZ-4D 


65 





3 


8 


2 


6 18 
6= -4D 


2 





3 
6» = 3 Z-2D 


7 





3 180 


3 


2 


6 18 

6» = 2F-2D 


2 





3 180 


2 





20 


2 


1 


3 273 


5 


1 


~ 3 87 
= Z-D 


27 


1 


3 93 

= ^ + D 


40 


1 


- 3 87 


2 



s 355 


3 


4 


9 


3 75 


16 


2 


5 


= «>3+2F 





75 













Motion of the Moon. 








35 ihb 






308. 


sv= 


= +o"-oo2 sin (2D + T — 3Q+ 105 ), 


Mercury. 




fif^^^^^H^n^^^l 






3°9- 


sv= 


+ o"-ooi C sin {0 +/M + i(M - T) + 


a }, Mars. 








6-- 


= 




6>= 2 D 




6> = 


= 2D-Z 




j" 


i 


a 


c 




j" i a c 




j" i 


a 


' ^^bEH 





I 





19 




-1-5 "49 3 




-5 


180 


'■m III 




2 




8 




- 2 - 6 297 2 




-4 




17 ^BlU 




3 




3 








-3 





5 HNII 




4 




2 




= Z 




- 2 




: : ■ HM 




5 




1 




- 1 180 2 




-1 -7 


149 


1 ll^^^^HI^H 


1 


o 


2I2'7 


5 




102 




-6 




^^HH 




i 




60 




4 180 3 




-5 




^■H 




2 


33 


10 




1 1 212 7 




-4 


3 2 9 


3 II^^^HIBIH 




3 




4 




2 32 1 




-3 




^^^^BS^H 




4 




2 


5 212 8 




-2 -6 


297 


^HIH 




5 




1 


- 1 - 2 147 1 










2 


i 


243 


2 




-1 327 7 




= 


2Z-2D 






2 




12 




2 2 243 1 




4 


180 


^^^■■S^l 




3 


63 


5 




6 63 6 




1 5 


211 


^HH 




4 




2 


- 2 - 2 297 1 




2 6 


243 


^^^■H^l 




5 




1 












3 


3 


276 


2 


6 = 2D- I 




6> = 


3Z-2D 






4 


96 


1 


0-6 180 1 ' 


2 6 


243 


^^HfEliH 






310. 


,57= 


+ 0" -oo i C sin { +/' J + i(J - T) + a 


°}, Jupiter 


• 






$■■ 


= 






(9= 2 D 






6 = 1 




f 


i 


a 


c 




/' i a G 




j" i 


a 


c ^BUH 





I 


O'O 


69 




-2 i8o'o 45 




1 


180-0 


■M HHH 




2 


180 


*3 




- I O 2 




2 




171 ^HHl 


I 


O 


i73'8 


209 




2 l8o 2 




3 


15 


2 b^^hss^ki 




I 


354 


11 




I O 174 2 




1 


173 


^^■Bl^Bi 




2 




8 




- I - 2 7 20 




1 


353'3 


2 H^^hH^bI 


2 


O 


162 


9 




O 2 




2 
3 


173 


58 Hill 




= 


2D 






= 2 




- 1 - 1 


186 


j HH» $181] 





-5 





1 




O-3O I 







6 


■ I KHNflBfl 




-'4 




2 




-2 4 




3 


286 


1 1 Hi 




-3 




3 




-1 8 




2 2 


342 


IB fill 

















sssass 



m*m 



36 



Mr. Ernest W. Brows, Theory of the 



§ V= + o"-ooi C sm{6+f,J + i( J - T) + a), Jupiter. 



= 2 D-Z 



= 4D-Z 



a 


(7 


4 180 


4 


3 183 


20 


2 iSo'o 


804 


1 


7 


3 260 


4 


2 3S3 


7 


4 187 


1 


3 187 


6 


2 67 


306 


1 280 


5 


2 18 


9 


1 107 


2 


e=2i) + i 




2 180 


3 


2 180 


1 


2 7 


1 






- 2 180 


7 


- 1 


-2 7 

B=2l 


3 





2 180 


*3 


1 


2 353 

#=2?-2D 


2 





2 iSo'o 


187 


1 


2 i73'3 


190 


2 


2 162 
= 2/-4D 


2 





2 


9 


1 


2 173 


6 



= 3/ -2D 






2 ISO 


7 


I 


2 173 
0=2F-2B 


5 





2 l8o 

0=2F + Z-2D 


2 





2 l8o 


2 



0= _ 2F + /-2D 

020 I 

1 o 81 4 






311. SV-= +0" -ooi C sm{6+f $ + 1(8 -T) + a°}, Saturn. 



90 



1: + 198° 
2V + 228° 
2 D 
2D - r + 338° 

1 + v + 192 



c 

24 



2 
O 

o 



180 

90 
90 



C 
3 
3 
3 



312, SV= +o //- ooiC sin >|/-, all planets. 






♦ 


14 


z - r + 168° 


4 


2D - 1 


17 


2D - l - V + 168 


2 


2 D + 1 


2 


2 D - 21 



:2 D-Z 





i 


a 


C 





- 2 


180 


14 


I 


- 2 


270 


4 



c 

2 

39 

1 
2 
2 















I^^^^^^H 






Motion of the Moon. 


37 HlM 


3*3- 


w= 


+ o"-ooi s in{0+/r + z(T-V) + a°}, Ferms. 


fif^^^^^H 


■ 


6= ±F 




(9= ±F + Z 


<9 = «;, 


§lf^^^^^fl| 




j i 


a 


y * c 


i * a 


a ft^^H 


^^H 


O I o 


5 


010 2 


O -7 285 


2 if^^^Hi 




2 


3 


2 1 


-6 


3 H^HI 




3 


1 


3 180 6 


-5 


^m 




i s 90 


2 




-4 


6 l^^Hl 




2 - 18 209 


2 


<9 = F + ?-2D 


-3 


9 fi^^Hl 








1 180 2 


- 2 


1 4 It^^^^BI 




<9 = F + 2D 




2 6 


- I 


27 i^Hl 




0-4 


2 


3 34 


I 105 


15 1 ■!! 




- 3 180 


1 


4 4 


2 


6 1 hH 




-2 


3 


5 2 


3 


3 i^^H 




- 1 


3 


1 5 90 4 


-2 -8 255 


2 if^^^^^K^n 




6= -F+2D 
0-8 180 

- 7 

- 6 

- 5 


2 
3 



9 


2 6 18 4 

= _F + Z-2D 
1 1S0 2 

2 6 

3 3 2 


-7 
-6 

"5 
-4 

-3 75'3 

— 2 


3 i^Hl 

9 I^H 

25 l^B 

72 l^Hl 

18 i^HI 




- 4 


2 3 


4 4 


- 1 


1 fl^^HI 




- 3 o'o 


45 


S 2 


1 


6 i^H 




- 2 

- 1 

- 1 - 6 90 


14 

9 

2 


2 6 18 4 
<9 = F-Z- 2 D 


2 
3 


3 l^H 




- 5 270-0 


68 


3 'i8o 2 


= W]-2D 






- 4 90 


2 


1 5 270 4 


~ 2 -3 255 


2 H^^BiS 




e=±F~i 




61= ±F+2Z- 2 D 


6> = w 1 ±Z 


Hi 




1 


2 


3 180 3 


- 2 - 3 75 


± 4 1 |H 




2 


1 


2 6 18 3 




If 




3 


14. 527 


= + o ;/, ooi C sin \J/- } Jupiter and Mars. 




* 


C : 


rf/ C . $ 


IBlisil 


±F + J + 34 8° 


2 


+ F + Z-2D + 2J-2T 36 


Wj + J + 69° 


^■1 1 ■'' 


F- 2D + 2J-2T+ 180° 


23 


±F + J-2D + 3J-2T+i73° 14 


F + 2D 


2 ff^^K»-.]| 




F-2D + 3J-3T 


5 


±F + 2 Z- 2 D + 2J-2T+i8o° 8 


-F + 2D 


1 HI $ 




- F - 2D + 2 J - 2T 


3 


±F + 2Z-2D + 3J-2T+ 173 9 


«; 1 - 2 T+75° 


8 B^il 




±F + Z+ 2 J- 2 T+i8o° 


7 


«ff 1 + 2M-T + 345° 10 








±F4-Z+ 3 J-2T+35 3 ° 


3 


ifj + T-J + 81 2 
















HP 

















3 8 



Mr. Ernest W. Brows, Theory of the 



315. § (Parallax) = + o ,/- ooi C cos ^. 



I - 2D + 3 T - 3V 

1-2V+ 2,1 - 2T 

Z-2D + 30 -2T+173 



C 
6 

7 
3 



* 


c 


i3T-8V+32i° 


3 


Z + 3 T-ioV + 33° 


35* 


Z+i6T-i8V+i5i°-o 


1455* 


Z4-29T- 26V+ii2°-o 


108 


I+21 (T-Y) 


3° 


2D- Z+21T- 2oV + 2 73°-0 


126 


2D-Z + 8T-i2V + 303° 


33 


2F-2D + 6I-5V + 270 


54 


3Z-2D + 24(T-V) 


10 


(! + 2F-4D-i5(T-V) 


2 


D+i2T-i5V+ 2 62° 


13 


D + 25T- 23V + 190° 


13 


3D-2F+19T- 18V + 272 


2 



316. Terms added to the elements. 
§w 1 = + o //- ooi C sin -\J/-. 



F + 24T-23V + 28S 

D-Z + F + 2o(T-V)+i66° 
D + ? -F+I7T-I8V+7S" 

3D - 3I + F + 2 5T - 2 2 V + 1 34° 

2 D-/ +5 T-4Q + ii3° 
2D-Z + T-3Q + 105 
2 r-Z + 3T-4Q + 67° 
3D-F-Z + 2T-3Q + 47' 

4D- 3Z + 25M — 23T + 67° 

D-F + 2M+165 
io q + no" 



8w 2 = + 2" - 69 (No. of years from 1850*0) 

+ o"'ii8 sin (1+ 16T- i8V+33i°-o) 

Sw 3 = - i""42 (No. of years from 1850-0) 

+ i""86 sin (w 3 + 29o° - i) + o"-i72 sin (/+ 16T- 18V+ i5i°'o) 





3 

2 
8 

2* 

3 

75 
3 
2 

4* 

17 

7 



8y= +o"'o83 cos (w 3 + iio't) 



Motion of the Moon 



39 



CHAPTER XII. 



THE INDIRECT ACTION OF THE PLANETS. 



Section (i). The Disturbing Function. 



317. Transformation to coordinates used in the direct action. — The disturbing 
function for the action of the Sun on the Moon is (Chap. I., § 3) 



% \ ft 3Z2 + tP 2 cos 2(V-V') + % £ cos 3 (V- V) + f (Tiz^) p cos ( V- V) j- . 



(1), 



to a sufficient approximation. Let Sr', SV be the perturbations of /, V from elliptic 
motion, the plane of reference being the same as before. Put Sr' jr' = Sp' and neoiect 
powers of Sp', SV 7 beyond the first. Then the disturbing function due to Sp', SV is 



R = J -^ 
4?- b 



^8p'{r 2 - 3 z 2 + 3p 2 cos 2(V- V')} +SF'{ 2/0 2 sin 2 (V- V')} 
•¥{#P 3 cos 3 (F- F) + (r2- 5? 2)p C os (F- V")} + 8 F{fp s sm 3 (V- V) + £(»•* - 5^ sin ( V - 7')}' 



(2) 



Replace the functions of the coordinates of the Moon and Sun by the expressions 
given in Chap. X., § 285, so that R will now denote that part of the disturbing function 
which depends on the lunar angle 0. We obtain 



ii = 



_ 3?» a- a 



Sp'(M 1 + |i¥ 2 ) cos 6 + 8 V'M S sin 6 - 8p'-,(M 6 + 5 M S ) cos + 8 F'?-,(jJ^ + \» M g ) 



(3). 



We can therefore obtain the required lunar functions directly from the results given 
111 § 292 if we multiply all the series there tabulated by aljr'. 

318. Final form of the disturbing function. — Denote the coefficients of these 
functions by accented letters when the multiplication a'/r' has been made, so that 6 
is now an angle in the products of the series of § 292 by a'/r'. Let an angle in Sp', 
8 V be (p, so that we have 

8p' = p c cos <f> , 8 V = -y, sin cj> , m' = n' 2 a' s . 

Putting a x for a/a' (§ 293), we obtain 

R = i«'%2( _ 3 )[( M ; + |M 2 ') Pc ± M s 'v, + ojCJ/g' + 5 Jf 8 ') Pc ± ^(£4// + V^s'K] cos (0 ± <p) . (4), 

which is in the required form (§ 279). 












40 Mr. Ernest W. Brown, Theory of the 

For terms p s sin <p in Sp', v e cos <ft in SV, replace p e by ±/> g , ±v s by v e , cos (Q±<p) by 
sin (0±(p). 

The values of fy/, 5 1 77 will be taken from Newcomb's tables of the Sun* (with some 
corrections). He tabulates io 9 log 10 (i + §p') = io 9 log 10 e . Sp'. If p e , p g denote his 
numbers, the parts in It which depend on these quantities must be multiplied by 
io" 9 log^io. The coefficients v e , v 8 are expressed in seconds of arc ; 1 shall consider them 
as expressed in units of o //- ooi, so that the parts of B which depend on v e> v, must 
be multiplied by io~ 3 /2o6265. The formulas of § 279, Chap. X., will then be available 
if we put m"/m' '- -j.ict 9 log e io= — [9*83934], so that 

/= -[4-13292], /= -[1-45682] ..... (5), 

and multiply v e , v s by the factor 

io^ s /2o6265 ^ io^ 9 log,. io = [32335] ..... (6). 

Then A is the portion of (4) within square brackets, after v s has been multiplied by 
this last factor. 

Section (ii). The Computation of Sp', SV. 

319. Forms of expression. — In this chapter perturbations of the first order 
relative to the masses of the disturbing bodies are alone retained. If we had used 
the method of the variation of arbitrary constants to find Sp\ SV, the variations of the 
six elements of the solar orbit would have been obtained in the form 

at + /8 + 2& COS (Xt + ix) , 

where a, /3, h, A, p. are constants ; in the coordinates we have a similar form, with the 
exception that in the elliptic terms k is of the form k't + h" and a = o in Sp'. Further, 
we can put a = ,8 = o in SV, All the periodic terms, except those which are indepen- 
dent of the argument of the disturbing planet, have therefore constant coefficients 
and are taken care of by the preceding method. Hence we have to consider only 
the terms 

Sp = M + 1(p { + tpi) cos (W + a.) , 

(*=i, 2,...) .... (7). 

8 V = 20',; + tv[) sin (W + a.;) , 

320. The non-periodic changes of the solar elements. — Now the solar eccentricity 
is an observational quantity, and we can therefore choose our arbitraries such that i\ = o. 
The other i\ and p,, p s are then so small that they may be neglected, and all the 
portions of the coefficients which depend on t may be expressed by a term e\t additional 
to e'. We have therefore only to add to the previous values of Sp' the term p^ cos (I' + %), 
which is treated in exactly the same way ; Set', which gives a constant term to R, 
and which is treated as in § 2 70 of Chap. X, ; and, finally, the effect of a variation 
of e'. The mean motion of the solar perigee is not quite zero, and therefore dl'/dt is 

* Amer. Eph. Papers, vol, vi., pt. 1. 



Motion of the Moon. 



4i 



not quite equal to dT/dt; the only term sensibly affected in the Moon's motion is that 
with argument V, for which the divisor n' instead of n' — dvs'jdt has been used. 

The treatment of the variations of e', xs' require special methods ; that of e' 
produces the well-known secular variations of w lt w % , tv s ; these have been many times 
computed, and their theoretical values are not in doubt so far as the lunar equations 
are concerned. It also produces terms of the form at + b in the coefficients of the 
periodic terms. These might be computed by means of the equations of variations, 
but I shall, in the next section, give another method which is much more simple for 
computation. 

321. Corrections to Newcomb' s values. — The values used in the solar tables 
(§ 318) are taken from his memoir* giving the computations. There are two sets of 
values in the memoir, obtained by independent computations, and the values of 
Leverrier are also given for comparison. These four sets of values were compared, and 
those in which the results agreed within the limits of accuracj 7 required were accepted. 
But certain of the coefficients (1) in which Newcomb and Leverrier did not agree, (2) 
in which Newcomb' s two sets of computations differed, and Leverrier's results were not 
given, (3) in which the degree of accuracy was not sufficiently high, or (4) in which 
the coefficients had not been obtained, have been recomputed. For this purpose the 
ordinary direct method was used — a method so well known f that it is unnecessary to 
do more than give the results ; these are included in the tables of Sect, (v) below. 
Nearly every one of the few errors found was typographical and easily detected. 

Newcomb has expressed doubts as to the sufficient accuracy of Sa' and the 
coefficients independent of the planetary arguments, and he has recomputed these 
portions. J I have thought it worth while also to recompute these parts by a modifica- 
tion of the direct method, shown in the following section, which gives the required 
formulas rapidly, instead of following the method of the variation of constants adopted 
by Newcomb. 

322. Computation of Sp' for the portion independent of the planetary argu- 
ments. — We shall only need terms of the second order with respect to the planetary 
eccentricities and inclination in the constant term, and terms of the first order in the 
coefficient of the principal elliptic term. In order to get the former, we do not need the 
second elliptic term, since it can only produce a non-periodic term in combination with 
a term of the same argument and therefore one of the fourth order. 

Dropping accents temporarily, we have for §r, 



T7 -(r8r) + n' i —r8r 
dt 2 ' r 3 



dR , 
--a-—- + 2«| 

da 



1 5ff? dt = (A + B cos I + C sin Z>i 2 a 2 
/ dw-. 



dw 1 



(8). 



where 11 is the disturbing function of the Earth's motion due to a planet, and I is the 
Earth's mean anomaly. All the letters except t are supposed to be accented, and 
A, B, C are quantities whose squares may be neglected. 

* Amer. Eph. Papers, vol. iii. t See, e.g., Cheyne's Planetary Theory, chap. vii. 

% Astron. Jour., No. 590. 

Royal Astron. Soc, Vol. LIX. 6 



mm 



42 Mr, Ehnbst W. Brown, Theory of the 

Putting a 3 /?- 3 = 1 + fe* 2 + yi cos I, we obtain by continued approximation for the 
particular integral corresponding to the terms on the right, 

'~=A(i-%e*) + %eCt + ±t(B-3Ae)&ml-±tC<ioal .... (9). 

No arbitraries are necessary, since they will disappear in connection with corresponding 
arbitraries in §V. 

323. The equation for the longitude is 

ncfi JT^SV- 2 4(»-§r)+ $>= - 2a j'^dt- 3 n j &df- = (D sin l + F cos l + Ft)na? , (10), 
^ at at J da J J dw 1 

suppose. Substituting for §r, we obtain, amongst others, terms of the form at cos I, 
fit sin I. These terms can be eliminated by supposing that rs, e receive increments 
Sm, Se proportional to the time ; as we are not computing these increments they may 
now be neglected. The constant term only adds to the observed value of the mean 
longitude for t = o; it may therefore be dropped. Let on, Se, §l be the changes 
necessary in n, e, I for t = o, in. order that the mean motion and the principal elliptic 
term may have same form as in undisturbed motion. 
Then 

(1 - |e 2 )8 V= t(F - \Be + %A& + hi ~ |e 2 8re) + cos l(E - C + 2eM l} ) + sin l(D + B- ^Ae + 2Se) . 

The coefficients of t, cos I, sin I, equated to zero, give on, Se, $l a . 
Finally, substituting in 

(1 + e 2 + 2 6 cos /) — - + — - Se cos I + eSl () sin I + JeSe , 

which is the total addition to log r (that is, the required Sp'), we find the terms 

A(i + e 2 ) + §F(i + %e 2 ) - ^Be - $De + ±(B + D) cos I + 1(0 -J£) sin I , , . (11). 

Let 

B = R Q + R c cos I + R s sin l = E + R 1 , 

and denote by J the operator ad/da. Then 



„™ .„...,. „., „ JR. .fdlt 



v?a?(A + Bcosl+0 sin I) = a-t" + 2 "-- dt = IR, + (I+ 2)R, , 

da J flWj 

m%%F+ D cosl-F sin I) = - 2a— - 3 l d h —dt = - zIR - (2/+ 3 )B, . 

da J dw 1 

The expression (11) becomes on substitution of these values 

w«a*[ -^(i-e2)/R + ^(7-1)5,-1(1+1)^ cos Z-K-?+i)B. Bin q. ■ • (12), 

a simple form which it is easy to compute. 

324. Let us now restore the accents and return to the usual notation. Then R 



Motion of the Moon 43 

becomes the m" j A of § 294. The terms required may be taken directly from Levekeuer's 
expression (§ 298). We have, for outer planets, 

RJkaa" = A4 ° + |(e' 2 + e^)(AA + AJ) - ly'ViA - U'e(AA +AJ- A A) cos (to' - ts") 

RjXaa" = - le!AA + \e'(AA - A J) cos (oT' - ct") , Ii s /Wt" = - fe"^ 1 - ^V) sin (tf - xs") , 

where 1 use the notation of Sect, (ii), Chap. XL, and certain relations* to reduce the 
expression for E ; also X = m"/'m'. The required formula for §p' becomes, on making 
use of the relation I(j> p = (p+ 1 )4> p+1 + f>(p p , satisfied by <p = A, (p = B, for outer planets, t 



S p ' = ,1-^'aa" { - UA + ^(2 A A - Af) - ^±^BA + ly'^BA + e AL(BA + AA- UA + ±AA) cos (W - ST") 
Til (4 ■■& 4 

+ r ^ a a" I ^(AA + A^) - e -(AA + } T AA - IAJ) cos (a' - To") 1 cos V 

+ ?Laa" \ 6 ^(AA + ±AA - Un 1 ) sin (T3 1 - rs") 1 sin I' . 

mi 2 ) 

For inner planets, E only requires the change a' for aa", but jR c , i?, are given by 

R e jXa' = l(^o + i«) - --(2^1 + AA) cos (gt' - ct") , R s /Xa' = ^{2 A A- + AA) sin (rs' - or") , 
22 2 

while in the expression (12) for Sp' we put — /— 1 for I. 

The values of a"A l p , a"B^, a' ' A' 9 , a'B l p are given in the auxiliary tables for the 
direct action. \ Those of the other quantities are found in Sect, (iv), Chap. XL The 
final results are included with the other terms in Sp', SV. 

Section (iii). Second Method, Application to Non-periodic Changes. 

325. Statement and Solution. § — The method may be regarded as a particular case 
of the genera] problem of four bodies, or of three bodies, or as a general method for 
treating any motion which is transmitted through one body to another, according to 
the view we wish to adopt. The last view will be that most convenient for our 
immediate purposes. 

Suppose that we have been able to solve, in terms of t and arbitrary constants, 
a dynamical problem which has a force function F. This function, expressed initially 
in terms of the coordinates, may also contain t explicitly and given constants. I shall 
suppose that it contains t explicitly only through certain functions of the given 
constants, u h , some of which may therefore be constant and some variable. Now 
suppose that, owing to some external agency, the u h are not the complete values of these 

* A.P.E., sect. iii. 

t The formulae do not quite agree with those of Nbwcomb given in Astr. Jour., No. 590, but the 
.umerical results agree with his as given in his paper, Cam. Inst. Publ., No. 72, p. 90. 
% A.P.E., sect. v. 
§ I have given the method in a paper in the Tram. Amer. Math. Soc, vol. vi. pp. 332-343. 



n 



WWWJW I WW I WI^ I li 



44 Mr. Ernest W, Brown, Theory of the 

given functions, but require certain additions, Su h , whose values in terms of the time are 
given. The ordinary method of treatment consists in substituting these new values in 
F and obtaining a disturbing function * 2 h (Su h dF/du h ) ; this is accounted for by find- 
ing what variable values must be given to the arbitraries, so that when these values are 
substituted instead of the constant values in the expressions for the coordinates and 
velocities, we shall have the complete solution of the problem. In this method no 
account is taken of the fact that F retains the same form with respect to the u k what- 
ever values may be given to these functions. 

In the memoir referred to, an attempt was made to utilise the absence of change 
in the form of F by considering the problem in the following way : — To find the 
variations of the arbitrary constants when not only their variations, but also those of the 
u h , are substituted in the expressions for the coordinates and velocities. I proved that 
if this plan were followed, the solution was equivalent to adding a disturbing function 

B-W^-tg), ...... ( I3 ), 

where the U h are defined by the differential equations 

dU h _ oF _, T1 3 du k dB . 

~tj ■ — 5 ■ 2 <k u k-^ — —;r - t. — j • • • • • ■ (14), 

at du h ou h dt ou h v ^' 

it being supposed that F is expressed in terms of the coordinates and the u h , that 
dujdt is expressible in terms of the u h , and that B is expressed as a function of the 
constants and of those u h which are independent 1 of t. 

326. Application to the secular changes of e', ts'. — We have initially e' , v>' con- 
stant. Let Sef = e{e't, Sxa' == vs^t. The u h are n', n't + e', ts\ e\ and therefore rs\ e ' are 
independent of the other u h and of dujdt. Also (loc. cit.) (J e ,, U w ' contain no non- 
periodic terms. Hence 

p '('OF,, ^.,[dF-,, 

S=- ei je (It-W, dt ...... (IS) , 



oe J oXS 



in which the non-periodic term arising from dFjde' must be -dropped. 

If we substitute this value of R in the equations of Chap. X., as we have no 
non-periodic part of R, §c ( = o, hv H = o, and therefore the secular accelerations are 
obtained by putting e'(i +e 1 't) for e' in the values of c v c 2 , c 3 expressed as functions 
of n, e, y, e', n', and finding the values of n, e, y which result, t The motion of rs' 
produces nothing in this connection, since it is not present in the c { . 

I have shown J that if we neglect af , a quantity which is quite insensible, the 
variations of n, e, y can also be obtained from the equations 

e.db„ , *db„ „ <> d f u \ 
c 2 d^ + c 3 8— ? = fS— (i- , ai = n,e,y (jd) 

* For simplicity, only variations of the first order are retained, but the methods are applicable when we 
take in higher powers. 

t This is Newcomb's theorem, Amur. Eph. Papers, vol. v., pt. 3, p. 191, 
% Proa. Lond. Math. Soc, vol. xxviii. p. 154. 



Motion of the Moon. 



45 



where (f*/r) denotes the non-periodic of fi/r , the functions being expressed in terms of 
n , e , y , e', n', the first four only receiving variations, that of e' being given. 

327. There remain the periodic terms of R. Since e\ vs' occur in R only through 
r', V, we have only to put the periodic terms 



Mr 

de 



■a, -: 



dp 



drs" 



SV-- 



. ,dV „,dV 



de 



dvs' ' ■ 



(17) 



in the formulae of §317, and, after rejecting all non-periodic terms, integrate; the 
resulting disturbing function consists only of periodic terms. 

The variations of the elements are then substituted in the coordinates. In 
accordance with the principle of the method, we must also put e'(i +e^t), m' + ra^t for 
e' vs' in order to obtain the true values of the coordinates. 

'- ; 7 

No other secular terms can be produced from the secular variations of the solar 
elements. It will be shown in Sect, (iv) that those of the inclination and node only 
produce periodic variations. 

The method of this section might have been used for all the indirect perturbations. 



Section (iv). The Motion of the Ecliptic. 

328. Choice of the Mean Ecliptic. — Owing to the action of the planets on the 
Earth, the plane of the Earth's orbit is not fixed, but has a motion which can be ex- 
pressed as secular and periodic variations of the inclination and longitude of the node 
with reference to some fixed plane. I choose as fixed plane the ecliptic at the date 
i850"o, and refer the motion of the Moon to the mean ecliptic at time t* The 
periodic perturbations are then included in the terms of the Sun's disturbing function 
which depend on z', portions which have been previously neglected (Chap. I., § 3). To 
be iu eluded in this mean ecliptic are one or two minute inequalities of very long period 
which then give rise to no terms in the Moon's motion, but which would do so if 
included in z'. 

329. The disturbing function for the moving ecliptic. — This is most easily found 
in a general manner. Let x, y, z, u, v, to be the coordinates and velocities of a particle 
of mass m, referred to rectangular axes which have velocities 6 1: 6 2 , 6 3 about themselves, 
and let the force function be denoted by mF. The equations of motion of m are 
then given by 



du_ 


oil 


dv_ 


dH 


(ho 


_oH 


lit 


ox 


dl~ 


dy 


~dt~ 


9.3 


dx 


dH 


dy _ 


dH 


dz 


m 


lit"' 


dll 


dt 


ov 


w 


ow 



where 

H=-^ + v 2 + te 2 )-F-B ! 

R = vz0 3 - wx6 2 + wy6 1 - uy6 s + uz6 2 - vz6 x , 

if we assume that 6 U 6. 2 , 6 S are independent of x, y, z, u, v, iv. 

* I have discussed this point in vol. lxviii. pp. 450-455, of the Monthly Notices, and have also given 
there the substance of this section. 



46 Mr. Ernest W. Brown, Theory of the 

When Ii = o, the equations become the same as those referred to fixed axes, and 
therefore li is the disturbing function for the motion of the axes. 

Let i! be the inclination of the moving ecliptic (xy plane) to that of 1850'Ci, t the 
longitude of the node on the fixed ecliptic, and for the origin of longitudes on the 
moving ecliptic take the "departure point" whose distance from the node on the 
moving ecliptic is the same as the distance from the node of the origin of reckoning on 
the fixed ecliptic. Then by Euler's equations 

„ di' dr ■ ., ■ n di . dr ■ „ a dr , ., N , „. 

1= = COST- SHI I Sltl T, P 2 = S1I1T + — sin* COST, 0, = —-(cos& - I) . (18). 

at dt at at at 

The values of i', di'/dt, dr/dt are small quantities of the first order, so that their 
squares and products may be neglected. On substituting the values, so limited, in li, 
we find the factor di'/dt common to all the terms, and therefore, since u, v, w differ 
from dxldt, dyjdt, dzjdt by quantities of the same order, 

di' r n~( dz /l!j\ ( dz dx 



di 



r , ( dz di/\ ( dz dx\ . , , 

tA = (y-7;-Z-T- COST- [X-—-Z-— suit .... (10). 

\ dt dtj \ dt dtj v Jl 



This disturbing function is available for any moving ecliptic so long as we 
may neglect the squares of its perturbations, but under the assumptions of § 328 
we substitute for i! only its secular part. The resulting disturbing function I 
denote by R t . 

330. The disturbing Junction for 'perturbations of the Earth out of the plane of 
reference is, if we neglect squares of z' and the terras dependent on a, 

E = -—:-^- ^d ...... (20), 

r " r * 

by § 3, Chap. I. With the notation and limitations of the previous section, 

z' = i'(l/' cos t - x sin t) . ... (21), 

Since this expression has the small factor i ', we consider t as a constant. In the paper 
referred to in § 328, I have shown that the expression (20) for li can be transformed 

into 

„ .,\( d % u d?z\ ( d?x ddz\ , \ ., dQ , N 

R = t 1 \z~- - y -^ cos t - \z~^ - x—; sm r > = - 1 ~^~ , . - (22), 

I V df df J \ dt 1 dt 2 J j dt ' 

which again is a perfectly general expression for the disturbing function when we can 
neglect squares of the perturbations. The value of Q is that given in the previous 
paragraph, and therefore the computation of one function serves for both disturbances. 
Under the assumptions of § 328 w r e substitute for i' in (22) only its periodic part, 
neglecting the minute perturbations of i' which are of long period relatively to 
that of the Moon's node. The disturbing function thus limited will be denoted 
by P H . 



i 

i 



Motion of the Moon. 
331. Computation of Q. — We h ave 



Q = ^psm(V-T)-z^L{p 8 m(V~r)} 



47 



= real part of (n-n')[Dz ■ wg-'e 4 "! 1 - zD(u£- 1 e w v)]e- 



" awg- 1 ■ D(a)-(D+i+m)(iz -u£fy 
k(i + m)a 2 



e( w i - T )' , 



The expansion of the portion in square brackets has been given in § 291, Chap. X. 
Let .1/ exp. 9 x i be one of the terms of this expansion. Then one of the terms of Q is 



given by 



Q = 11AM sin (6 X + w l - t) = nAM cos ($ 1 + s 



332. Computation of R v — Let di'jdt=p . Then 



n ' m 



pi'VM cos [6 1 + w 1 -t- 



7T\ 

V. 



and -therefore if we put m" 'jm! for the factor outside the square brackets the equations 
of § 279, Chap. X., can be immediately applied ; here A = M. 

The value of p is* d'-^jio per annum, n'=3548 //- 2 per diem, and therefore 

4('47io)k r- n 



giving 



m (36s'25)(3548-2)m, 

/=[6' 2 333L /' = [i'SS72]- 



In finding derivatives with respect to n, we must use d(nka, 2 A)/dn. 

333. Computation of R t . — Let P cos (p be any term of i' reckoned in units of 
o /,- ooi, and s x the number of seconds in the daily motion of the angle Q l + w l — t = 6. 
Then, using the first form for Q, we have 



where 



7? =fLi 
m 



t' 2 -d 2 A cos (6 ± <f>) 



A=- ^MP , 



k , r— 



m' 1000 206265 3S48'2 m 



— - = +[12-5149] . 



The computation of derivatives with respect to n must be made with SjAfa 2 . 
The last fact makes, the computation easier with formulas (20), (21). We have 

B 2 = fn'H'^J pe {sm(V-r)- sin (V- 2V + r)} 

= \riW-, { - M'P cos (0 ± <£} with h" = t , 

+ |w'% 2 ^{ Jf 4 "P cos (6 ± 4,)} with r = 2F-r, 

using the third formula of § 285, Chap. X., after multiplication by a'jr'. For these 
* Leveebibr, Ann. Ohs. Paris, vol. iv. p. 50, after correction for the adopted masses of the planets. 



48 Mr. Ernest W. Brown, Theory of the 

terms it is sufficiently accurate to neglect the solar eccentricity in finding the Moon 
factors ; this gives M\ = M'\ — M A , F ; = T. The value of m"jm' is 

K= ?- .—^=[5-16273. 

m 1000 206265 

The terms, arising from these disturbing functions, in the results at the end of 
this chapter, are those in longitude which contain the argument w s explicitly, and 
those in latitude which contain the argument w 1 explicitly. The most important are 
two of the latter with arguments Wjdt^T — 3Y) and having coefficients o"'OJJ and 
o //- 030 respectively. 

Section (v). Numerical Values of the Earth's Perturbations. 

334. Sources. — The general values for op', §¥' are taken from Newcomb's tables 
of the Sun (§ 318), with some corrections and additions (§ 321) ; the secular variations 
of e', ra' are from the same source. The terms independent of the planet's arguments 
are found in § 324. The values of i', r are from Leverrier,* after correction for the 
masses of the planets adopted here ; the coefficient of 2J — 2t in i' was recomputed. 

335. Notation. — The values of §p', SV are given by Newcomb as cosines and sines 
of the mean anomalies ; it was convenient to retain them in this form. Leverrier 
gives the values of i' in terms of literal arguments ; these were combined and expressed 
in terms of the mean longitudes, which is the final form for all the inequalities due to 
planetary action (Chap. XV.). Hence, I put 

$, = £ cos {f(T - rf) +j"(T - vs")} + £ sin {/(T - To') +i"(T" - tr/')} , 1 = ! c cos (j'T +/T" + «') . 

Also, v e , v g are expressed in units of o""OOi, while p a p s are expressed in absolute 
units multiplied by io 9 Jog 10 e, that is, they are the coefficients in io 9 log 10 (i +Sp). 
For the secular terms, de' jolt = e/e', d&'/dt — vsr 1 . 

* Ann. Ohs. Paris, vol. iv. p. 50. • 



Motion of the Moon. 



49 



336. Venus, T" = V. 



,/ 

o 
I 

2 
O 
I 

2 

3 
o 
1 

3 
4 

2 

3 

4 

5 
6 

3 

4 
5 
6 

7 
5 
6 

7 
8 
6 

7 
8 

9 



12 

J 3 
14 
16 



1 8 



Pc 




P* 




Vs 




Ve 


+ 627 














+ 14 


- 


7 










- 85 


- 


39 


- 


67 


+ 


33 


- 2062 


_ 


1 1 46 


~ 


4228 


+ 2353 


+ 68 


_ 


14 


- 


34 


- 


65 


+ 14 


- 


8 


_ 


8 


- 


3 





+ 


4 


+ 


i 


- 


3 


+ 84 


.!_ 


136 


+ 


60 


.. 


99 


+ 3593 


+ 


5822 


+ 2903 


- 


4702 


- 59 6 


- 


632 


- 


*737 


+ 


1795 


+ 40 


+ 


33 


- 


33 


+ 


3° 





+ 


21 


+ 


1 


- 


13 


+ 44 


+ IO44 


+ 


27 


- 


666 


- 381 


- 


1448 


- 


397 


+ 1508 


+ 126 


+ 


148 


- 


684 


+ 


763 


+ 14 


+ 


13 


- 


12 


+ 


12 





+ 


6 


_ 


1 


- 





- 166 


+ 


337 


- 


93 


_ 


188 


- 5i 


+ 


189 


- 


38 


- 


J 39 


- 25 


- 


9 1 


- 


42 


+ 


146 


+ 3 


+ 


5 


- 


4 


+ 


5 


- 134 


+ 


93 


- 


69 


- 


47 


- 39 


+ 


43 


- 


2 5 


- 


28 


- 37 


+ 


136 


- 


33 


- 


119 


+ 0-3 


- 


27-9 


- 


1 


+ 


154 


- 80 


+ 


8 


- 


38 


- 


4 


- 24 


~U 


7 


- 


13 


- 


4 


10 


-f 


10 


- 


7 


- 


6 


+ 3 


- 


12 


+ 


3 


+ 


14 


- 38 


- 


17 


- 


18 


+ 


8 


- 14 


- 


19 


_ 


7 


+ 


9 


- 43" 2 


+ 


8-i 


- 


41 


- 


8 


+ 9'oo 


- 


7-71 


+ 


416 


+ 1251 


- 25-2 


+ 


22-3 


+ 


24 


+ 


21 



24 



24 



3° 



43 



39 



209 
180 
209 



209 



209'4 



29 



299 



see § 350. 



Royal Astron. Soc, Vol LIX. 



rnmmm 



5° 



Mr. Ernest W, Brown, Theory of the 



337. Mars, T" = M. 



o 

I 

- 2 

- I 
O 

- 3 

- 2 

- 1 

o 

- 3 

- 2 

- 4 

- 3 

- 2 

- 1 

- 4 

- 3 

- 5 

- 4 

- 3 

- 5 

- 4 

- 6 

- 5 

- 4 

- 6 

- 5 

- 7 

- 6 

- 7 

- 6 

- 8 

_ 7 

- 9 

- 8 

- 9 



10 
11 
13 

15 

i7 



Pc 




fs 


Va 


- 13 








+ 5 


— 


3 




5 


+ 


6 


4 


- 92 


+ 


119 


- 167 


+ 27 


- 


6 


- 47 


- 13 




5° 


10 


- 573 




[976 


- 567 


4- 64 


- 


i37 


- 617 


- 18 


- 


25 


+ 15 


- 154 


- 


67 


- 118 


- 77 


- 


201 


- 153 


+ 46 


- 


17 


+ 3 2 


+ 461 


+ 


125 


+ 483 


+ 43 


+ 


96 


- 256 


+ 6 


+ 


8 


5 


+ 87 


- 


62 


+ 69 


+ 87 


+ 


17 


+ 200 


3 


+ 


30 


2 


- 102 


+ 


94 


- 113 


- 27 


- 


4 


+ 100 


+ 4 


+ 


60 


+ 3 


- 26 


+ 


28 


- 72 


12 


- 


9 


8 


8 


- 


44 


- 10 


+ S 


_ 


6 


— 12 


- 3° 


- 


16 


- 25 


4 


- 


17 


I 5 


+ 7 


_ 


3 


+ 5 


+ 14 


+ 


6 


+ 18 


+ 17 


- 


10 


+ 15 


+ 8 


+ 


3 


+ 42 


1 


+ 


!5 


— 1 


4 


+ 


3 


- 33 


- 17 


- 


14 


- 16 


0-90 


- 


5-92 


- 3* 


1 '3 


- 


o-6 


+ 24 



- 216 



The last figure was not computed, and is 



+ 


4c 


+ J 


963 


— ] 


659 


- 


24 


+ 


53 


+ 


396 


+ 


11 


- 


131 


+ 


526 


+ 


7 


+ 


49 


- 


38 


_ 


20 


- 


104 


_ 


11 


_ 


49 


- 


78 


+ 


6 


+ 


5i 


- 


17 


+ 


13 


+ 


60 


+ 


2 


- 


7 


+ 


9 


- 


12 


- 


13 


- 


3° 


+ 


13 


4- 


20* 


_ 


10 


not 


needed 



263 



Hiiilii 



Mm;,,,-, .,/' i i,,' M.„,„. 



5' 



338. Jupiter, T" = J. 



Jilllil§ll 


Hilt 




Pc 


' f* ■ ' : 


• "' .•• ■ *s ''/'■■ ' 


''■■'.' 'Vc, .';•• . :■•';.. 


it- 


: 


<; 


•iJ: ; 3 •;■••; - ; 


513 










IllSlllllfl 




#111^18%^ 


39 


■ ■ +' * :; ?2 •"'.';• 








~ 5 


1 


*V*V ; ' •- 


'2 •••';■• 


+ 5 


— I 


3 




_ -> 




- 


78 


+ 193 


- 52 


■'. ' : ~ >55 " ; 




illilllllli 




;■•:>'. ■ >.;+'* 


56 


+ 7067 


+ 59 


- 7208 




e 




-.- ; ■ : • + 


.22-7'. / 


- 89 


-2582 


- 3&7 


- 6 


iltlltlllfl 




+ 


79 ' : 


+ 9 


''=-■• 73 . 


.: '. +.-'.'8 , ', 




lltifill 


8111 


+ 


.lb. a: \:;. 


- 17 


+ 68 


- '+ -i 1 • • 




- _• 




. +.4021 •..•'• 


- 203 


:-±- 21<Z&:<: 


+ 136-. .- 


'3'3 


Slllllllll 




1 > +' 


[376 


+ 486 


' +IS18 ' 


'•'• -' -537/ . 




^llllllll^llii 




- 


I 


8 


. - -70... 


— 22 


'S 2 


Pllfll 


3 


.;■ V/ : .' . -4" 


43 ; 


+ 278 


• '+.' 27" ; 


- 162 




J 




M"-': '":''-¥ 


796 


— 104 


+ 55' 


"'.'.+ . 7i 




lllllillllt 




''■'■ ■■''.- '.*' : * 


172 


• + 26 , 


+ ••2.08 


- 3' 















... 




18 


- 4 


4 


^'."•"''■ : ' ; ~- 


29 


+ • .5 


- 16 


3 




- 3 




7.; ;- ; -V : -*' 


13 


• +• 73, V 


+ 9 


. - "43 '. 




- 2 




; -" : .- * : '.' ; + 


no 


- iA- ■ 


■+ v 78 ' ■ 


- 17 




- 1 




,> : 


17 


+ • 1'.. ' 


+ 23 


- 1 





354 



1 1.) 2 



I 5° 



^39. Saturn, T" = S. 



p- \ 





'■•• . ■""'.: 


24 






















2 


1 


'/'■':.■ + 


. J 5 . '• 


' +•;• 


.; 




:,,*' 


II 


- 


3 


34i- 


.-!// 


f/fililrls 


I 




.4 


422 : 


' ;'+'. 


79 




"•■•'+• 


412 


- 


77 


Secular 


terms. 


O 




' : V. 


7-89 


• : ~" 


O' 


53 


- 


320 


- 


3 








1 




~ ' -'+' ; 


8 




: ° 




- 


8 : ; 







S'tt'j' 


= ; p- '..' 




- 


2/ 


W-:-/-^-:' 


1 5 ^ 


W.'SSv'i: 


57 




- 


101 


'••/+•' 


38 


sr/ 


= +[6 


"9537]"' 


1 




- ""^ 


103 


■ ■ '—y 


44 




■ ; ~! 


103 


• + 


45 


S» 


— o'.. 









; . '+'• 


0-31 


' ' — 


• - 0: 


56 


- 


*7 . 


+ 


-2 


«]' 


--P 


"59.68]«' 


.; 


3 


:-.>"- ' ■ 


30 ' ; 


\ ; ^:j^-i 


11 




■^-r::"^ 


20 


•.• +:, 


7 


T 


= 173° 


•46 


1 




— 


16 


/■vrr--^^ 


6 




^; : x^ 


16 


+ 


6 


"f 


--- +[7-5604].// 



340. Mercury, T" = Q. 



27 

5 

2 



+ 



- '3 



19 



Mr. Erxest AY. Brown, Theory of the 



Section (vi). A Sieve fere the Rejection of Insensible Coefficients. 

342. Terms to be considered. — Just as with the direct inequalities (Sect, (iii), 
Chap. XL), we only need to consider the possible terms of long period. Also, as 
before, I consider separately those that do or do not contain v: x in their arguments. 
The latter can be at once dealt with ; the number of possible terms in the lunar 
factors is practically limited to those with arguments 2D — 2I, 2D — 2F, 2 F — 2 /, I) — I, 
and in any case it is a brief matter to consider the possible combinations with the 
planetary arguments of all but 2D — 2/, 2D — 2F, which arc computed in the regular 
course with the short-period terms. 

Those that contain iv 1 in their arguments can only produce sensible terms in 
combination with planetary factors having nearly the same period, that is, a month. 
Hence we have only to consider the values of coefficients in op', SV belonging to 
terms which hare periods of about a month or less. 

343. Construction of the Sieve for terms containing u\. — The equation for cV 

is (§ 322) 

d- ,'«,-., n-o'P , s , .flR' , , fdR' 7 , , 

-jror) + ——rar==a--^+2nl--—c/t (21), 

r ■• da J d t ' " 



dP 



where R! = rn'jA ; the other portion of M' gives coefficients which are quite insensible 
in the class of terms considered here. Let q be the mean motion of a short-period 
argument in R', and therefore in Sr : for the periods approximating to a month 
in eV we have rlq = m approximately. 

As far as the effect of the periods is concerned, the four terms of (23) are of 
relative orders <r, n n , n' t , 211'q. And further, the largest terms in R' with periods 
approximating to a mouth must have high multiples of T — T" in their arguments, 
and in this case the two derivatives of R' are of the same order of magnitude. 
Hence the order of So' for such terms is given bv 



11 -a 'bp 



, d.R' rP 

■a . -,- _ - or 



ila 



c q- 



da 



The equation for SV may be written 

, , ,- — ,- a 



at 



,(SV): 



n'ci", ,. , a u ,dr 
2 ---'W^ br) ~ad^dU 



{24). 



dli' 

irF 



dt 



(25). 



Similar reasoning shows that the order of the right-hand member is the order of 
its last term, so that, on integration, SV is seen to be of the same order of 
magnitude as Sp'. 

Now A 2 = r'' i + r' 2 — 2r'r"<r, where a- is the cosine of the anode between r', r". 
Hence 

.f'lTl . fill ../ -1-"^ ~~ ■■}•'' ^ i \ r m'' r/' z 

. . (26). 



"'iliii' ~"dr 



-— = order of 



for the worst case, that of Venus. Hence 3^// is of order (iii"ii)i')(a' : PA H )m 



Motion of the Moon. 



DO 



Take first the inequalities depending mainly on Sf. A comparison of the 
disturbing functions in §§ 294, 517 shows that the order for the indirect terms 
is to that for the direct terms as $Sp : (m/ / fm')(«/ 3 l& s ), that is, m 2 : 1, since we 
mav take 3V/ A 6 as having the same order at the worst as i/A 3 - Hence any term 
which is shown by calculation to have a coefficient less than \" due to direct action 
will not be sensible in the indirect action. There is only one term left, that with 
argument 1 + 16T- 18 V, coefficient i4 /; '55 ; the order of the coefficient for the indirect 
action is this number multiplied by m\ that is, the order o"'oo. (the computed 
coefficient found below is o //, o6). 

The terms due to SV are treated in exactly the same way and give similar 
results. The direct inequalities are all small, and there are no sensible ones arisino- 
from the indirect action. 

Hence, there are no sensible terms of long period containing the argument tv 1 
mid arising from indirect action in the plane of reference, except a small term 
having the argument of the great Venus inequality. 

344. Terms arising from i'. — The principal argument in the Moon factors is 
m>, and the combinations of this with the comparatively few terms in 1' which are 
sensible are first studied ; then the terms of one order higher with respect to the lunar 
eccentricity and inclination, and so on. It soon becomes quite obvious that the 
only terms beyond those of lowest order in the Moon factors must be of long 
period relatively to that of the Moon's node. As the terms in i/A to be considered 
must have the factor y !,i , their number is very limited. 

The methodical search for long-period terms was simple. The tables formed for 
the periods in the sieve for the direct terms were available.* As w B is itself of lone 
period, it was only necessary to combine w z with terms in that table which contained 
multiples of w. 2 — w s , w 3 which were either both even or both odd, that is, terms 
for which the multiples of w. A or F were even. The combinations of these with w 3 
were those required. Only two survived, apart from those with the sole lunar 
argument iv s , namely, the arguments 



2W 9 + ?<> 3 - 2.J = 



W - w B ■ 



2D - 2I- w 3 - 2 J + 2T, period 277 years ; 
T, period 540 years. 



2,J = 2F-D-Z+2i- 3 -2j 



lire latter is quite small compared with the former, on account of the lunar characteristic 
k"a,e as compared with e 2 k, and the planetary factor is also much smaller ; the period 
is twice as long, but this only multiplies the relative coefficient by 4. 

In a note at the end of a paper lately published, t I gave a value for the 
coefficient of the former term as o"-2\. Since the paper was published the term 
ims been recomputed with the disturbing function (20) ; this revealed an error in 
trie former computation, and the coefficient appears to be d'-oo^. It is therefore 
not retained in the final results. 



A.P.E., sect, v. 



Monthly Notices, vol. lxviii. p. 170. 



mwm 



54 Mr. Ernest W. Brown, Theory of the 

Section (vii). Computation of the Lunar Perturbations. 

345. The disturbing functions are given in !$§ 318, 324. 327, 332, 333, the 
values of the planetary factors in the last section, and those of the lunar factors in 
Sect, (v) of Chap. X. The method of arrangement was to take one lunar argument 
with all multiples of T — w' and form the products for all the planetary arguments. 
The equations of variations have been so arranged that the process of finding the values 
of hvi, on, Sc 2 , Sc 3 from the disturbing functions is very brief and simple, inasmuch as it 
was rarely necessary to compute more than two of these six variations. In fact, in the 
few cases where more than two were needed, a simple ratio, the same for all terms, 
generally sufficed ; such a ratio was also sufficient in the majority of cases to find all the 
other variations after the principal one had been obtained. 

The experience gained in computing the direct inequalities suggested that the work 
could be much abbreviated by considering the peculiarities of each lunar argument, and 
these peculiarities are set forth in the following paragraphs. 

346. The primaries independent of the lunar angles. — Here §n = Sc z = Sc 3 — o, 
and Stv 1 is first computed, and then - eSw 2 , so that the secondary arising from the sub- 
stitution of hv. 2 in the principal elliptic term, - 2e cos I. Sw 2 , is obtained directly. For 
$w s , it was sufficiently accurate to treat §w s /8w 2 as a constant which is the same for all 
terms, and indeed for the small terms, kv 2 : Sw 3 : hv 1 are constant ratios with sufficient 
accuracy. 

When these primaries and secondaries have been found, the remaining secondaries 
can be written down almost by inspection, I therefore only give the coefficients of the 
primaries arguments \l, and of the secondaries arguments \1/±Z. 

The primaries of very long period arising from terms of very long period in $p', 
§V are treated in § 352 below. 

347. Primaries containing w x and independent ofu\, w 3 .— Here the periods of all 
the terms are very nearly the same, Sw 2 is nearly equal to Sw t , and Sw s is about }3w 2 . 
Hence the secondaries are all very small, the largest being less than o /; -oio. All of them 
greater than o"'ooi were computed, and will be included in the final results, but it is 
unnecessary to print in detail any but the primaries. The principal variation is 
again Sw v 

348. Primaries containing w 1} w 2 or w z only. — These are the terms in which the 
secondaries are generally much more important than the primaries, and in which the 
theorem of § 282, Chap. X., has its full force, the principal variations being $w, 2 , Sc 2 . If 
the primary contain w. z in the form il + <p (i positive), the principal secondary is that 
with argument (i -j)l + (j>, and this was first computed in all cases ; then the value of 
Sw 1 ; §w B was insensible, or only produced very small terms. The terms divide into two 
classes according as their periods approximate to a month and less, or are much longer. 

The first class includes the lunar arguments 

J, 2D -I, 2D + I, 4D-Z, 2/, 



Motion of the Moon. 



53 



for which the principal terms, those in longitude, have the lunar arguments 

o, 2D, 2 D, 4 D, I, 
respectively. 

The second class contains only one argument, 2D- 2I, for which the principal term 
in longitude has the argument 2D — I. 

I give Sw t , that is, the primary, and the coefficients of the principal set of terms in 
longitude only ; from these all others can be obtained immediately. 

349. Primaries containing iv x , w s or w s only. — The principal variations being Sw 3 , 
Sc s , the largest term is always the principal secondary in latitude, and the statements of 
the previous section can be repeated. But only two lunar arguments have to be con- 
sidered : 2F, which belongs to the first class, and 2F-2D, which belongs to the second 
class. The principal terms in latitude contain the lunar arguments F, F - 2D, and the 
coefficients of these four terms are alone set down. 

350. Special terms. — -Beyond the inequalities mentioned above, there are two with 

arguments 

D-Z-4T + 3V, Z + 16T-18V, 

and with periods of 94 years and 273 years. The former is computed in the same 
manner as the other terms. For the latter we require to find §p', SV for the argument 
16T- 18V or x6(T-ts') - i8(V -vs"). It was known (§ 343) that the final coefficient 
was of the order of o //- i. It was therefore sufficient to find the terms with multiples 
16, 17, 18 of T — tff' combined with i8(V -m"), so that they might be combined with 
the lunar arguments I, l-(Y-xa'), l—2(T-xs') in order to give the required argument. 
The direct method of computation gave (in absolute units), with sufficient accuracy, 

8p' 



•0000059 00s 






i6(T - &') - i8(V - ui") + 129° 



8V - '0000046 sin ( 
"hence arises the only sensible term, 

§•«/'!= - o"'o6 sin (1+ 16T- 18V+ 150° 



f 



m 
m 



(27)- 



351. Omitted terms.— The portions of the disturbing function independent of 
the planet's mean motion are the constant term and those with argument V . The 
former chiefly produce small changes in the mean motions of the perigee and node, and 
also slight changes in n, e, y which affect the coefficients of the evection and variation, but 
the latter are quite insensible. The variation of e' produces the secular accelerations. 

The terms due to the motion of the ecliptic and the latitude of the Sun do not 
combine with any others due to Sp', §V. 

The detailed results given in the next section do not contain these terms, nor the 
terms in the two following paragraphs ; they will all be given in the final results due to 
indirect action contained in the last section of this chapter. 

352. Terms left as perturbations of the elements. — A few of the primaries 
which have very long periods and which are independent of the lunar angles are 



■m 



56 Mr. Ernest W, Brown, Theory of the 

so left, and their values are to be found with the collected results at the end of 
this chapter. The principal inequality due to the motion of the mean ecliptic is 
also treated in this way. 

But there are short-period primaries due to long-period terms in Sp', SV which 
are sensible. I shall show that these may be accounted for by the substitution of 
w-l 4- long-period terms in SV for w{ in the final expressions for the coordinates 
of the Moon. 

Consider how such terms would be treated by the second method in Sect. (iii). 
The chief perturbations of the solar coordinates arise from Sn' and Sw x ' = jSnfdt + Se. 
Hence we should have Su x = Sit', Su 2 — Sip/, so that 

d 5, *du, d * ' d <* , ^dw,' rl,-, , 
— ou, — d — l = on , -610, — o — l -=--oe. 

dt ' dt dt dt x at at 

Now- hi', Se only contain the first power of the period as a large multiplier, and 
therefore the disturbing function does not contain it at all.* Hence all the terms 
produced by the use of this disturbing function are quite insensible. 

We therefore proceed as with the secular accelerations, obtaining the variations 
of n, e, y by solving the equations csc t - = o considered as functions of n, e, y, n' in 
which Sn' is known. Then §w 1 — \hidt, etc. But this simply gives the primaries 
independent of the lunar angles, together with the secondaries arising from them, 
and these we have already obtained. The method therefore proves that for all 
other terms arising from long-period terms in SV, Sp' w T e are to simply substitute 
the disturbed values of ,rt', w-[ in the final expressions for the Moon's coordinates. 
The substitution of Sn' gives nothing sensible. Hence the statement. 

353. Computation of the secular variations. — The values of Sn, Sb 2 , Sb s have 
been obtained in an earlier memoir. + With the adopted value of Se', namely, 
e/=-[6-5968]< I find 

8n= + [Ti'o294.]rm't , S& 2 = - [7i - 8476]mm'£ , 8b 3 = +[Ji'o'j46]nn't , 
giving 

8wj = + 5"-8 242 , 8iv 2 = - &"•&* , Sw 8 = + 6"- 4 6^ , 

where t c is the number of centuries from 1 850*0. 
To obtain the values of Se, Sy, we have 

o = 8c„ = ?p Sn + c iS Se + *p Sy + d ff8e , 
an de dy de 

with a similar equation for c 3 . As Sn is known, the values of Se, Sy can be found 
from the formulae (39) of Chap. X.. by putting for Sa 2 , Sc s the expressions 

idc„ ,, ide<, ,, 

- e df e ^' ^di e ^- 

We find that Se/e. Sy/y are less than io~H C) and therefore quite insensible. 

* This is true in general. See Trans. Amer. Math. Soc, vol. vi. p. 341. 
t Monthly Notices, vol. lvii. pp. 342-349. 



Motion of the Moon. 



57 



354. Description of the tables. — The portions of the detailed results selected 
for printing have been described in the preceding paragraphs ; it is understood that 
all portions greater than o"*ooi have been computed and included in the collected 
results o-jven in Section (viii) below. 

The heading to each page sufficiently describes the contents in general It will 
be noticed that \j/ + a° always denotes the argument of the primary, \f^±/ + a° being 
the argument of the principal secondary in longitude, and ^±F + a° that in latitude. 
The angle rs" is the longitude of perihelion of the particular planet considered, and 
V the mean anomaly of the Earth's orbit. 

a is the same for all values of C in a given line until a new value is 



The angle 



set down. 



Royal Astron. Soc, Vol. LIX. 



58 



Mr. Ernest W. Brown, Theory of the 



3 5 5 . S V = + o"-oo i C sin { i'V + i"( V - ro") + + a} = + o"-oo i sin (^ + a), Venus. 









e = o 






a= 2 D 




6i= - 


2JJ 






* 




*:W 




+ 


■ \ 


* 




i' 


i" 


a 


G 


a 


C 


a 


c 


a 


c 


o 


- I 


334 


8 


33 2 


2 


152 


10 






i 




330-9 


354 


33°*9 


72 


150-9 


III 


150-9 


50 


2 




48 


12 


90 


3 


319 


2 


175 


4 


3 




28 


1 














i 


- 2 


122 


13 


122 


5 


302 


I? 






2 




1 2 1 '6 


5" 


121-5 


104 


3°i'5 


179 


3° 2 


11 


3 




3i5'4 


275 


3 2 4 


39 


i33'7 


54 


134 


30 


4 




3 2 3 


5 










136 


3 


2 


-3 


91 


1 






272 


2 






3 




g2'o 


50 


91 


10 


272 


23 


91 


; 


4 




284-8 


158 


285 


S 2 


104-7 


46 


104 


8 


5 




3 r 4'5 


129 


322 


22 


i3 2 


14 


I3 1 


20 


6 




315 


2 










133 


2 


4 


-4 


64 


14 


64 


3 . 


244 


9 






S 




74 


13 


73 


3 


255 


5 






6 




287 


17 


288 


3 


106 


4 


106 


1 


5 


-5 


35 


5 






215 


4 






6 




48 


3 






226 


2 






7 




74 


12 


74 


3 


253 


4 






8 




271 


19 


274 


4 


90 


3 


90 


2 


6 


-6 


6 


2 






186 


2 






9 




256 


2 














7 


-7 










156 


1 







Motion of the Moon. 



59 



§V= +o n -ooiCsm{i , l , + i"(Y-W) + e + a°}= +o"-ooiCsm(^ + a°). 



) = l 




i 

2 
1 

l 

■■ 

■• 
4 



-4 



a 


V 


a 


332 


1 




33°'9 


13 




296 






122 


2 




121-5 


24 




335 


5 


-&■ 


91 

285 

3°9 


3 

6 

2 


O 

CS 


64 


1 




73 






287 






35 






74 






271 









4 

47 

1 

7 
92 
18 

13 

21 

5 
5 
3 
2 
2 
2 
1 



-I 



$ + 1 



33i 

5 

302 
3i4 
3*7 

2 73 
103 
312 
244 



21 

7 
1 

4 
3 
7 
i 









<9 = 


2D + Z 


(9 = 


-2D -1 


6» = 4D-^ 


<?= - 4 D + 2 








* 


i|/-J 


* 


t+i 


i/» ty + l 


+ -Z 


t? 


i' 


a 


c 


c 


c 


c 


C G 


(7 





- 1 


!5 2 




2 










I 




151 


4 


20 


2 


10 


2 5 


2 


I 


- 2 


302 




3 






1 




2 




301 


6 


32 




2 


2 7 




3 




*33 


2 


10 


1 


6 


2 


I 


3 


~3 


271 




4 










4 




105 


2 


8 




1 


2 




S 




x 33 




3 




4 






4 


-4 


244 




2 











6o 



Mr. Ernest W. Brown, Theory of the 



SV= + o"-ooiCain{i'l' + i"(V -w") + + a 3 } = + o"-ooi Csin (| + «1. 









6=2l)-l 




0= -2D + Z 










* 


* + z 




<l> 


♦ -* 


i' 


i" 


a 


c 


c 


a 


c 


c 





- i 


332 


4 


19 








i 




330-9 


41 


214 


330-9 


16 


91 


2 




300 




2 


358 


2 


9 


I 


- 2 


121 


7 


32 








2 




121-4 


72 


359 


122 


5 


13 


3 




313*3 


!9 


100 


3*3'3 


9 


55 


4 










316 


1 


6 


2 


-3 


92 




S 








3 




91-0 


10 


47 


272 




3 


4 




284-7 


19 


91 


284 


2 


J 3 


S 




3 12 


5 


26 


311 


7 


37 


6 










3i3 




3 


4 


-4 


64 


4 


20 


244 




2 


5 




74 


2 


10 








6 




286 


1 


7 


286 




2 


S 


-s 


35 


2 


9 








6 




S 2 




4 








7 




74 


2 


8 








8 




271 


1 


5 


271 




4 


6 


-6 


6 


1 


5 








7 




18 




1 








7 


-7 


336 




3 








8 


-8 


3°7 




2 













0= 


= 2l 


0= -2? 


i' 


i" 


a 





(7 


1 


— 1 


151 


3 


I 


2 


- 2 


3 02 


5 




3 




134 


1 




3 


-3 


272 


1 





Motion of the Moon. 



61 



S V, SU=+o"-ooiC sin {i'V + i'"( V - ra") + 6 + a] = + o"-oo i C sin (^ + «°). 



:2 Z- 2 D 



o 
i 

2 

I 
2 

3 
4 
3 
4 
5 
6 

4 

5 
6 

5 
6 

7 
8 
8 



-4 



= - 2l+ 2D 





* 


*-z 




* 


i^ + Z 


a 


c 


c 


a 


c 


c 


159 




4 


33 2 




5 


33°'9 


3 


77 


33°"9 


4 


87 


2 




3 


108 




1 


122 






122 




6 


301 


6 


IO 


121-5 


6 


112 


3i4 


2 


36 


3 r 4 


2 


48 


3*7 




2 








92 - S 


-3 


59 


91 




12 


284 




11 


285 


2 


33 


311 




19 


312 




16 


3i3 




1 








64 




4 


64 




4 


2 54 




3 


74 




3 


286 




2 


286 




3 


36 




1 


35 




1 


45 




1 


74 




2 


270 




2 


270 




3 


236 




4 









- I 

- 2 
-3 



331 

122 

3 r 4 

284 
311 



)=2F-2D 

c 

I 

I 



C 

11 

6 



33i 
121 

3i3 

gi 

285 

313 



-2F4-2D 




* 


* + F 





C 


I 


11 


2 


14 




7 




1 




4 




2 



Mr. Ernest W, Brown, Theory of the 



35 6 - 



S Vr= + o"-oo i C sin {i'V + i"(M - w") + + , 



■ o" - ooi O sin (\f-- + a"), Mars. 











#=0 








$=2D 






| (9= -2 


D 








+ 




*±z 






* 






! * 




i' 


i" 


a 







■ « 


C 


a 




c 




1 a 





- I 


I 


S3 




3° 


53 


7 


232 




/ 




232 


3 







9 




5 














196 


1 


-3 


2 


285 




4 


285 


1 


106 




6 








- 2 




286-1 




206 


286-1 


42 


107-2 




61 




105 


10 


- I 




72-8 




271 


81-4 


7i 


251 




29 




249 


3 2 


O 




299 




3 














265 


3 


-3 


3 


335 




n 


335 


3 


151 




5 








- 2 




291-6 




5° 


292 


11 


in 




10 




1 11 


5 


-4 


4 


194 




3 


194 


1 


5 




2 








~ 3 




164-8 




5 1 


165 


11 


347 




15 




342 


3 


- 2 




2 97'3 




82 


297 


18 


117 




9 




"5 


1 1 


— I 




3°3 




1 


















-4 


S 


215 




7 


215 


1 


3i 




3 








-3 




170 




2 5 


170 


5 


349 




5 


; 


349 


2 


-5 


6 












254 




1 








-4 




42 




16 


42 


3 


223 




4 


| 






-3 




176 




*4 


176 


3 


356 




1 




35° 


2 


-5 


7 


93 




5 


93 


1 i 


270 




2 


1 






-4 




48 




'3 


48 


3 


226 




2 




228 


1 


-5 


8 


280 




6 




1 


IOI 




1 


! 






-6 


9 


332 




3 




6 = 1 


15° 
+ -1 




1 
= 


-I 

4- 


ty + l 










i' 


i" 


a 


c 


C 


ce 




c 













— I 


I 


52 




3 


















-3 


2 


285 




2 


















— 2 




287 


8 


29 


104 






5 










— 1 




67 


2 


8 


69 




3 


11 










— 3 


3 


33 2 




2 


















- 2 




292 


1 


4 


















-3 


4 


166 


2 


7 


340 






1 










- 2 




293 




2 


296 




1 


4 










-4 


s 


213 




2 


















-3 




170 




2 


















-4 


6 


43 




2 













Motion of the Moon, 



S V= + o"-ooi C sin {i'V + i"(M - m ") + 8 + a°\ =+ o"-ooi C sin ty + a"). 







6 = 


2D-Z 




61 = 


= -2D + I 










* 


i^ + 2 




* 


4,-1 


i' 


t 


a 


C 


C 


a 


c 


C 


— 2 


i 


S3 




1 








- I 




S 2 


* 


12 


5 2 




5 


O 










is 




2 


"3 


2 


286 


2 


10 








- 2 




287-0 


2 3 


117 


285 


3 


15 


— I 




70-4 


10 


5 2 


68-8 


11 


57 


O 










86 




5 


-3 


3 


33i 


2 


10 








- 2 




291 


3 


18 


291 


1 


8 


-4 


4 


186 




4 








-3 




166 


5 


27 


164 




4 


- 2 




297 


3 


15 


2 95 


4 


20 


- I 










299 




2 


-4 


5 


212 


1 


6 








-3 




169 


2 


9 


169 




4 


-5 


6 


74 




2 








-4 




43 


2 


8 


42 




1 


-3 




i7S 




2 


172 




4 


-s 


7 


90 




3 








-4 




47 




4 


48 




2 


-5 


8 


282 




3 








-6 


9 


3 2 9 




2 













(5 = 


= 2D + Z 




0-= 


-2D-Z 


<9= 4 D-/ 


e= -4D + 1 








+ 


+ -Z 


* 


if- + Z 


<|< + z 


,;,- 


if 


t" 


a 


c 


c 


c 


c 


C 


c 


- I 


I 


232 




I 










_ , 


2 


105 




I 










— 2 




106 


2 


II 




2 


2 




- I 




i°5 


1 


5 


1 


5 


1 


I 


— 2 


3 


in 




2 












4 


347 




3 










~ 2 




117 




2 




2 







III 








ǤPP 






W^W:8'' 






8||t; : sp::'' 






^K^-^-Bt*' 






Hly ^j'- - y •* : : 






flSffpS^ 






SK£ : w: : : : Sx : : 


















M& : : : :* fg : : 






IS ; I'll 




bwb? r ^fH 








Biii 




BESjfey : :%s : : : : : gl« 









S» : i 



1 



6 4 



Mr. Ernest W. Brown, Theory of the 



§V, SU= + o"-ooiCsin {i'l' + i"(M-za") + 6 + a°}= +o' , -ooiCsin (j + a°). 



i' 


i" 


= 

a 


2^-2D 




c 


= 

a 


- 21+ 

c 


2D 

c 


- I 


I 


52 




3 


5 2 




6 


o 




14 




1 








-3 


2 


287 




2 


286 




2 


- 2 




286 




12 


286-9 


3 


42 


- I 




6 9 


1 


32 


70 


2 


29 







90 




2 








-3 


3 








332 




3 


- 2 




291 




6 


291 




9 


-4 


4 


197 




3 


186 




1 


-3 




165 




3 


166 




10 


- 2 




295 




11 


296 




9 


-4 


5 








212 




2 


-3 




169 




3 


169 




4 


-5 


6 


74 


- 2 


5 








-4 




42 




1 


43 




3 


-3 




173 




2 


174 




1 


-5 


7 








91 . 




■ 


-4 




48 




2 


47 




2 


-6 


8 


*35 


— 1 


3 








-5 










281 




1 







0=2F- 


2D 


e= - 


2F + 


2D 


e= 2 i 






*-F 






fr + F 




*-i 


i' 


i" 


a 


c 


a 




c 


a C 


— 2 


2 


286 


3 


286 




6 


I06 2 


- 1 




67 


4 


70 




4 




-3 


4 






166 




1 




- 2 




2 95 


1 


2 97 




1 





Motion of the Moon. 



65 



jD/' 



sv= 



+ o""ooi 



C sin - f i'V + t"(l - w") + + a } = + o"-ooi C sin (^ + «°), 
Jupiter. 



6 = 



$±i 



a 


C 


a 


77 


17 


82 


9°'4 


724 


9C4 


348-4 


262 


3 I2-6 


2 


8 


359 


186 


. 


185 


1827 


208 


182-5 


161 '6 


r 73 


163-0 


358 


14 


348 


102 


9 


105 


187-0 


43 


186 


173 


23 


174 


101 


3 




191 


7 


191 


'77 


3 





45 
39 
6 
2 
9 
5 





6 


= 2 


I) 




a 








C 


265 








t-9 


269*7 








211 


182 








39 


5 








10 


2 *2 








104 


339'9 








44 


183 








2 


290 
7 








7 
21 



351 

287 

It 



271 

190-5 
182 



35 2 



2 1) 



34 
46 

5 

2 
9 









6 = 1 






= - - / 






i" 


a 


c 


(7 


a 





c 


2 


1 


83 


2 


8 








I 




90-0 


27 


102 


2 72 




14 







17 


4 


13 


8 


5 


16 


I 















2 


3 


2 


185 


1 


4 








2 




182-4 


14 


56 


3 


2 


14 


1: 




162 


6 


2 3 


344 




3 


3 


3 


107 


1 


4 


281 




1 


2 




187 


3 


1 1 


7 




3 


1 




173 




3 








3 


4 


104 




1 








2 




191 




1 









Royal Astron, Soc, Vol. LIX. 



■■ 



66 



Mr. Ernest W. Brown, Theory of the 



S V= + o"-ooi C sin { i'V + i"( : ) - rs") + + a' / ] = + o"-oo i C sin (\|/ + a). 









0=2D-Z 




61 = 


-2D 


+ / 










* 


■+ + « 




* 




•J--J 




i" 


a 


c 


C 


a 


c 




C 


2 


I 


84 


7 


37 










I 




89-8 


82 


417 


90-9 


10 




58 







4'5 


' 14 


75 


IO - I 


16 




87 


3 


2 


185 


4 


19 










2 




182*3 


44 


214 


3' 2 


1 




6 


I 




1 60 - 3 


17 


89 


160 


3 




IS 


O 




8 




3 


44 






3 


3 


3 


109 


3 


IS 


276 






1 


2 




187-2 


9 


42 


6 






1 


I 




171 


2 


11 


179 






2 


4 


4 


18 




2 










3 




106 




4 










2 




192 


1 


6 










I 




177 




1 











- 2 

- I 
O 

^3 

- 2 

_ j 

~3 

- 2 

- 1 



265 
270 
181 

S 
2 

34° 
291 

7 

35* 



2D + ^ 




* 


*-l 





C 




4 


7 


39 


1 


7 




2 


4 


19 


1 


8 




1 




4 



)= - 2 D-Z 



272 
191 



340 



* 


*+<: 


c 


c 


I 


7 


2 


9 







<? = 


4 D- 


-/ 
•J- 


<(/ + <: 


6 = 


^ 4 D + ^ 


\jl-l 




i" 


a 




C 


e 


a 


c 


c 


- 2 


I 


267 






1 








- 1 




269 




3 


9 


272 




I 







l8l 






2 


J 93 




2 


- 2 


2 


2 




1 


4 








- 1 




340 






2 









Motion of the Moon. 



6 7 



:F= + o"'ooi C sin {i'V + i"{3 - vs") + 9 + a°\= +o /, -ooi C sin (^ + a). 









e=2i~ 


2 1) 




6 = 


- zl+ 2 


I) 










* 




ty-i 




* 




t|< + £ 


i' 


i" 


a 


c 




c 


a 


<7 




C 


- 2 


I 


72 






9 


84 






8 


- I 




90-6 






47 


89-9 


8 




iS° 


O 




8-9 


2 




46 


6-2 


2 




44 


I 




1 






3 










^3 


2 


187 






1 


185 






3 


- 2 




182-8 


53 




338 


i82'4 


3 




53 


- I 




1S8 






12 


161 


2 




33 


O 




33 






1 


iS 






1 


-3 


3 


100 






2 


108 






3 


- 2 




S'9 


94 




130 


187 






11 


- J 




170 






2 


172 






4 


- 2 


4 


11 


1 




4 


192 






2 









6=2l 










•J- 


4--Z 


% 


i" 


a 


C 


a 


- 1 


1 


270 




6 







187 




1 


- 2 


2 


3 




3 


- 1 




34i 




1 



6= -2? 

4- i|< + 2 



C 



C 









e = 


= 2F- 


2D 


4--F 




= 


-2F 


+ 


2D 


>J< + F 


i' 


i" 


a 




c 




C 


a 




c 






a 


- 2 


1 












84 










1 


_ 1 




91 




1 




10 


90 




2 






19 







11 








6 


2 










6 


- 2 


2 












182 










7 


1 




161 








2 


l60 










4 


- 2 


3 












187 










1 



68 



Air, Ernest W. Brown, Theory of the 



35 8. SV= +o"-ooi(7 sin {i'l' + i"(S-ia") + e + a c 

Saturn. 



= +o"-ooi(7siu (4r + a), 









+ 






f±/ 






G = 2B 


(9 = 


- 2 1) 


' 


i" 


a 







a 




C 


a 


c 




c 


I 


I 


169-4 




42 


169 




8 


35° 


12 




2 


o 




3S 6 -s 




45 


354 




16 


180 


5 




5 


2 


2 


339 




8 


339 




1 


'59 


4 






I 




337 




n 


33? 




3 


156 


3 











117 




3 
















2 


3 


340 




2 
















I 




341 




2 























= 1 


e= -1 


0-2D-/ 


6= -2D 








i i-l 


4* + 1 


i, f+l 


# 




i" 


a 


a c 


c 


G C 


a (7 


2 


1 


169 






2 




1 




169 


6 


- I 


5 24 


169 







O 


2 


2 


2 10 


8 2 


2 


2 


339 


2 




2 9 




1 




337 


2 




1 6 




2 


3 


34° 






2 










6>= 2 D + I 


(9= -2D + Z 


6i= 2 Z- 2D 


0= - 2I + 2D 








*-z 


*-l 


+ -* 


* + ? 




i' 


i" a 


c 


c 


a C 


C 




- 1 


I 349 


2 




169 2 


II 







I 80 


I 


I 


6 


6 




.... 2 


2 






341 5 


2 




— 1 








337 


2 




- 2 


3 






334 1 





*-J 



359, «f F= + o"-oo6 sin {V + 347 ), a« planets. 



Motion of the Moon. 
Section (viii). Final Results for the Indirect Action. 



69 



:6o 



>. Tests.— No general method of testing the work appeared to be available, but 
various peculiarities of the solution very much aided in the avoidance of the kind of 
error winch is most likely to occur — one running through a whole series of terms. 

In order to try and abbreviate the work, the two terms with the same argument 
of the form a sin \f^ + & cos ^ in the disturbing function were combined into a sino-le 
argument of the form A cos (^ + «) as early as possible. This might have become 
very disadvantageous in the final process, where terms with different values of A, « had 
to be added together. It was, however, obvious that for nearly all terms with the 
same ^, the angle a should be nearly the same ; and the cases where this was not to 
be expected were evident. This approximate equality of the angles therefore served 
as a test. 

Again, let us consider the terms with arguments independent of the lunar angles. 
They are formed of the primaries with arguments <p and the secondaries with arguments 
(</> + 1) - l,(<t> - 1) + 1, (<p + 2D - 1) - 2D + 1, etc. When the final addition was made, it 
was found that the sum of the secondaries was always small compared with the 
primary, unless the primary was a term of very long period— an exception of rare 
occurrence and easily noticed. Consideration of the peculiarities of the method of 
variation of the constants showed that this must necessarily be the case.* This fact 
furnished a full test of the principal terms whose primaries have the arguments (p±l, 
since these secondaries are the largest terms arising from those arguments ; it was also 
a partial test of the terms with arguments $±(2!) -I) in the disturbing function. 

The final terms containing cp + 2D arise chiefly from the primaries with arguments 
<p + 2 1) and the secondaries with arguments (<£ + 2D - 1) + 1. In general, the latter are 
very nearly half the former ; this can also be shown to be a consequence of the theory. 
This tests the terms with arguments 2D, 2D - 1 in the disturbing function. 

The only important terms not tested by these methods are those with arguments 
</>±(2Z-2D), all of which have periods much longer than the month. The resulting 
terms in the longitude which have the largest coefficients are those with arguments 

0±(/-2D). 

L'He existence of these tests raises a doubt as to whether the variation of arbitrary 
constants is the best method for treating the numerous short-period terms. Possibly a 
direct method might be more simple. It would certainly have the advantage of finding 
the changes in the coordinates directly, and of avoiding the formation of derivatives 
with respect to n. It might be advisable to use a direct method for most of the terms, 
and to use the variation of the elements only for those of long period. 

361. Arrangement.— The various terms with the same period which arise from 
f ie substitution of the elements in the Moon's coordinates have been collected into one 
term ; in some cases there were as many as eight such terms. 

* See Monthly Notices, vol. lxviii. p. 166. 



M 













jo Mr. Ernest W. Brown, Theory of the 

As with the direct inequalities, the terms are divided into classes, those added to 
the coordinates and those added to the elements. The selection is somewhat arbitrary, 
and it largely depends on convenience for the formation of the tables of the Moon's 
motion which will be a natural sequence of this work. There are no terms which seem 
to require the other class used with the direct terms — those partly added to the co- 
ordinates and partly to the elements. In other respects the arrangement is quite the 
same. 

The angles in the preceding section were the mean anomalies of the planets ; they 
are changed here to mean longitudes to facilitate addition with the direct terms. 

The terms given are those equal to or greater than o"-ooi, with the exception of a 
few (e.g. in the latitude due to Mars) for which there are no portions due to other 
causes. But all terms greater than o //- oo2 are included, except one or two of very lone- 
period, noted above. 

In the terms added to the elements, t e represents the number of centuries reckoned 
from 1850T). 

The two terms added to n't + e are to be included in the arguments D = nt + e- n't — e, 
and l f = n't + e' — vs'; and the secular part of w' is to be included in the latter argument. 

All terms given at the end of Chap. IX. which contain the arguments dtil' (1 positive) 
are to receive the factor 1 +i'Se'/e' = 1 — •00248*7,,. 



I 

2 

3 

4 

s 

6 

- 2 

- I 
I 
2 

3 
4 

S 
i 

2 

3 
4 

5 
3 
5 
6 



Motion of the Moon. 



562. <5F= +o // -ooi(7sin {9+fF + i(T -Y) + a c '}, Venus. 



O'O 

179-8 
179-6 
180 



259 

77 
323 
273'3 
271-8 

90 

93 
216 

180 

20I'I 
20 3 
*9 

92 
115 

120 



)=2D 



c 

344 

5°7 
5° 
14 

S 
1 

14 



[ 55 
14 

3 
1 

5 

129 

18 

12 

2 
19 

2 



4 


180 


2 


3 





2 


2 





10 


1 


180 


38 


1 


O'O 


84 


2 


179-6 


144 


3 


178 


17 


4 


180 


8 


S 




5 


6 




3 


7 




2 


8 




2 




232 


3 


2 


271 


40 







(9= 2D 




^ 


a 


1 


3 


271 




4 


89 




5 


103 




6 


92 


1 


-3 

- 2 


269 




— 1 


254 




1 


101 




2 


28! 




3 




2 


3 


199 




4 


202 




5 


20 


2 


-4 


338 




-3 


342 




- 2 


7 


3 


S 


116 


3 


^5 


64 




~3 


91 



-4 

~3 

- 2 

- 1 
1 
2 

4 

- 2 

- 1 
1 

2 

3 

4 

-4 

-3 



c 


i 


i 


37 


- 1 


- 1 


S 




1 


2 




2 


1 


2 


3 


6 




4 


23 




5 


5 


- 2 


-5 


8 




-4 


13 




-3 


3 


3 


5 


11 


— 3 


^5 



3 
4 
1 

3 

2 
2 
1 



u u 







8 




90 


i8o'o 


61 


O'O 


61 


i8o'o 


112 




4 


258 


4 


79 


1 


27 


3 


271 


33 


272 


34 


9i 


3 


89 


3 


268 


27 


269 


33 



-5 

^4 

-3 

- 2 

- 1 
1 

2 

3 
4 

5 
6 

- 2 

_ ] 
I 

2 

3 

4 

-5 
-4 

3 

- 2 

- 1 
1 



151 
102 
282 
209 
204 

19 
161 

336 

33 1 
115 

65 



)=2T>-1 



C 
3 
3 
7 

20 

3 

3 

3 

3 

21 

3 

3 



O'O 


1 




4 




58 


i8o'o 


IS 




87 


O'O 


120 


179-6 


167 


178 


21 


180 


8 




3 




1 


259 


3 


252 


4 


37 


1 


271-4 


65 


271-9 


49 


90 


5 


90 


1 


270 


3 


269 


10 


268-1 


43 


2 5i 


S 


101 


9 


281 


13 



■UK 
Up 


E 




3gSm£>: 


$§:-:-:*!i 






: 8h«P&"'- 


Mii 






llPi 


* II 




?:£&: 






•:-: ; : : :"> 




Bl| 






iBSI 


(£>:$ 

w 





Mi Kiwim W. |;i:r\\ n /'/./.„_,/ ,,/' '-' 



- r 



3 
4 

5 

- 6 

:-'4- 

3 

_■ 

5 

" 5 

— 3 



)■-■-. -A > 

200 
202 

•9 

34-- 
3.^ 

34° 
6 

i 14 
66 
91 



i=2D + / 



.,:>"•, >-.,i( 'sin ; f» +./T + .'iT- \'i + . ( ;. !'.-,,».<. 



.», 

4 
4 
•i 

2 
24 
3 
4 
2 
j 



I 


180 


4 


1 


: ' : ;'i//.V'6 ;:V-' 


10 


-• 


]So 


'5 


3 




3 


4 




1 


2 


271 


5 



t 3 

- 1. 2 

1 

2 

2 3 

~ 2 -3 



271. 
269 
IOI 
28l 
.' \ 199 

v -•: 341 ; 
(J-4D-/ 



1 = 2/. 



2 


-73 


3 


3 


272 


2 


~~ 3 


268 


2 


-2 ..' ; 


207 


3 


3 


.::.: 


1 


-3 


-3* 


1 



1 


180 


6 




(9 = 


^ 2/ - 2D 




2 
3 



92 

e_=2i 


7 
1 





2 

— I 
i 
_» 

3 




180 




7 
5 

'\- 2 
8 

3 


2 





7 


1 


2 


2 7 2 


2 


- 1 


180 


5 


— 1 


"3 


211S 


3 


1 





5 




- 2 




3 


2 


180 


8 


2 


6 


200 


3 



<5F= +o //, ooi(7siii(0 + 0). 



- 4 T- 


3V + 


272°'I 


<^ = 5T-3V + 2i6° 


8 







.: • /' '»..'/• • " ' ; - ' " 


/- l> 




51 


.. ' '«3 ' ^ X 9 


-I) 




3 2 


w a ±l 3 


2/- I) 




2 


w 8 + 2F 4 


1 ' - 1 s 


2° 


7 


-w g -2F 2 


-/-- J) 




3 





563. SV= + o"-ooiG sin { 6 +/'M + i(M - T) + «°} , Mars. 



& = o 



/' 


i 


a 


c 


j" 


i 


a 


"',-:"' X^. 


/' 


t 


a 


C 





1. 


180 


30 


1 





36 


s 


2 





183 


2 




.' 


1 So - 2 


203 




.1 


2 2 ■;•■•; 


269 




2 


*45*° 


Si 




3 


35'> 


1 1 




2 


2125 


48 




3 




25 




4 


34-' 


3 




3 




5 2 




4 


244 


16 


[ 


■3 


2 5o 


6 




4 


320 


9 




5 


62 


5 



Motion of the Moon. 



73 



SV=+ o"-ooi C sin { 6 +j"M + t(M - T) + «°} , Mars. 



i 


a 


C 


/' 


f 


a 


c 


I 


iS° 


i 


-3 


-3 


264 


2 


3 


277 


14 




- 1 


34 


2 


4 


276 


14 










5 
6 


275 
94 


6 
3 






6 = 1 





0= 2 D 



- 2 





5 


- I 




2 


I 


180 


4 


2 


181 


44 


3 





5 


4 


328 


2 


1 


224 


23 


2 


212 


6 


3 


214 


8 


4 


37 


3 


5 


16 


1 


-3 


327 


1 


- 2 


328 


3 


- 1 


3i7 


23 





139 


1 


2 


280 


1 


3 




3 


2 


244 


5 


3 




4 


4 


246 


4 


5 


59 


i 


-4 


296 


1 


-3 




2 


- 2 


297 


8 





40 


2 



3 3 275 

4 276 

5 277 

6 90 
-3-4 264 

Royal Astron. 



1 
2 
2 
1 
1 
Soc. 



0=2D 



-3 


180 


3 


- 2 





38 


— I 




6 


I 


180 


7 


2 




43 


3 





3 


-3 


261 


1 


1 


224'2 


66 


2 


212 


11 


3 


213 


13 


4 


30 


1 


S 


I96 


1 


-4 


ISO 


1 


-3 


33° 


9 


- 2 


327 


9 


- 1 


304-2 


67 


3 


279 


1 


2 


246 


16 


3 


245 


5 


4 


244 


3 


5 




1 


-5 


296 


1 


-4 




3 


-3 


29S 


S 


- 2 


294 


17 


3 


277 


3 


4 


276 


3 


-4 


264 


3 


-3 


263 


2 



o -4 

- 2 

, Vol. LIX. 



2D -2 

19S 

o 



3 
11 



> = 2 D-Z 



i 


a 





1 


O 


3 


1 


180 


8 


2 


181-0 


61 


3 


353 


5 


4 


329 


1 


3 


259 


2 


1 


220 


31 


2 


212 


11 


3 


214 


14 


4 


27 


3 


5 


164 


5 


3 


327 


2 


2 


328 


6 


1 


320 


35 





139 


1 


3 


280 


4 


2 


244 


11 


3 




6 


4 


245 


5 


5 


60 


1 


6 


3°9 


3 


4 


296 


1 


3 




3 


2 


297 


14 


3 


275 


1 


4 




2 


5 


276 


1 


4 


264 


2 


3 


266 


2 



> = 2 D4-Z 

180 
o 

, 82 

212 
214 

93 



6 
I 

3 

1 

2 
3 



10 



mini 



74 



Mr. Ki:n i>t \\". I!i:.)\v\ : '/'/„;. ,-,/ -,/ ,'/,. 



SV= + o"-ooi C sin {6 +/'H + i(M - T) + «°}, Mars. 



!=?D + J 



?=2? 



^2?-2D 





«" o ' ' 


■ 


i" 


W0^~iM: 


;;i:^ : ::^y::^:\y::i 


c 


•/'• 


•t 


a 


C 


2 


2 245 


. 1 





- 2 


9 '. 


3 





— 2 


O 


4 


2 


— 2 297 


1 




■ 2 


ISO 


3 


1 


1 


87 


1 








1 


I 


232 


3 




5 


196 


.: 




<9 = 4D-£ 


■/■/■^^■::}:y; : ::^ 


- 1 


- I 


308 


3 


- 1 


- 1 


98 


.■; 


O 


2 


2 










2 


6 


51 


1 



364. S V= + o"-ooi Csm{8 +fj + i(,J - T) + «°} , Jupiter. 

2D 6 = 1 



i: 


;vK;|;s:«j§sj!;|| 


C 


I 


178-9 , 


712 


2 


'359*6 


200 


3 


7 


10 


-3 


.. 2 57 : 


6 


-2 


'274 


18 





33< v 3 


259 


1 


238-0 


170 


2 


35 2 


44 


3 


355 


4 


- 1 


25° 


10 





334 


14 


1 


238 


2 5 


2 


344 


6 


1 


230 


3 



) = 2D 



■3 





: I 


- 2 


180 


25 


- I 


1 


31 


I 


i 7 s- 5 


J.67 


2 


359'2 


87 


3 


r 3 


. 7 


4 . 


12 


2 





3 19 


29 


1 


237 


35 


2 


352 


*5 


3 


358 


2 



i 


a 





- 2 


. 9 


10 


- 1 


303 


6 


O : ' 


184 . 


35 


'2 '• 


273 


9 


3 


102 


.6 . 





. 35i 


'i'' 


1 


236 


5 


2 


345 


3 


- i 


288 


1 '• 





200 


. 3 


1 


no 


6 


1 


230 


1 



)=l 



-3 


173 


• ■-% 


- 2 


180 


40 


- 1 


i*o 


136 


1 


179-0 


i5° 


2 ' ' • 


180 


19 


3 


.25 


3 


- 2 


274 


6 


* 0' : ." 


298-5 


71 


1 


239 


40 


2 


35 1 


38 


3 


• 257 


i 


- 2 


188 


7 


- 1 


301 


36 



3" 


* 


:;:«;:; 


C 


- 1 . 





• 24i's 


:- 




2 


273 


6 


' -2 ' 


— I 


250 


2 







326 


7 




I 


238 


5 




.".2 


344 


2 


- 2 


— 2 . 


196 


.! 




— r 


302 


5 







214 


7 




1 • 


290 


2 



= 2D-/ 



-3 


*75 


2 


— 2 .. . 


180-4 


333 


- I 


1 


■4-4- 


I 


178-4 


2 11 


2 


359'2 


s 9 


3 


14 


6 


-3 


z6 3 


1 


;_ :2' ; 


279 


9 


• :-P 


• S'S' 


56 


I 


237-0 


46 


2 


352 


20 


-2 ' . 


9-1 


130 


_ I" '. ' 


301 


'3 


O : 


174-2 


60 


2 


273 


16 







Motion of the Moon. 




75 1 


j:^:lljjl|j:^^;:;:IE::; ■■ 














S V= + o"-oo i C sin {0 +/'J + i(J - T) + a°} , Jupiter. 




»:| 1 




6=2l)-L 


e = 2l) + i 


8=2l-2~D 






j" 


i, a 


j" i a. 


j" i a 


1 


~ : HHilll1IL-::: : : : fri>>; : <??: : S 


- I 


3 102 7 


-12 273 3 


0-2 180 


5 1 


y:H»ffl££:3-{g:2:-:*: 1 


2 


o 35i i 


3 102 I 


- I 2 


11 MB 


^SUf; &' 1 I 




i 237 6 




I O 


3 S 




- 2 


2 344 3 
-2 196 4 


1 358 9 


2 1797 

i 2 I 7°"9 


53 I 
94 » 






- 1 305 2 


2 179 5 


- 1 - 1 302 


2 HI 


IBKllill 




171 1 


1 172 1 


186 


2 fig 


|:aHH& 1 r § -JH 




1 291 3 


1 57 1 
— 10 2 1 


2 2 164 

6>=2Z- 4 r) 


I ■ 


1 Hk ' ! Jh 




e=2i) + i 


(9=2/ 


1 2 171 


1 Mi 


:.-MKflBre : >: £$:•: ]3»fl 


O 


-1 1 5 


O -2 180 3 










1 178 21 


- I 2 II 


<9=2F- 2 D 




^Ssliift:^ l« 3S 




2 359 8 


I 178 12 


- 1 


2 H 






3 r 4 1 


203 


1 180 


1 ■ 




I 


353 4 

1 237 s 


'10 293 5 

I 239 3 






ffii : « 




2 35 2 3 


2 171 I 


# = « 3 




m^^^^K^^-'-^»^bB 


- I 


- 1 3°3 1 


-I -I 301 3 


10 


3 1 


Sra«3»§<^i»- : -:-i ^9aH 




182 4 


O 247 5 


2 168 


6 I 


«HBBWBk>:?¥x-:': 3K&H 




365. cfF= +o"-ooiCsin {6+j"S + i(S- 


T) + a°[ , Saturn. 




flippy 19 




6 = 


<9=2D 


e=2D-i 




ililpspi IH 


i" 


i a C 


j" i a C 


j" i <* 


c H 


BSlSSKg: * #£|:o 3rH: 


o 


i 179 6 42 


1 I 257 3 


O -2 l8o 


5 H 




208 


2 270 2 


- I 


2 '((■■f 


If iff- 1 ■ iS 


I 


266'6 45 


-IO 255 4 


I 180 


14 H 


IMH9 




1 257 13 


2 280 2 


2 O 


4 H: 


iltl 'it 




2 270 2 




I O 4271 


6 Hi 


j|J|t < £ ; IB 


2 


297 3 


6 = 1 


2 257 


3 ■si 


H&Kffifi:! S&iHf 




i 171 2 


-2 180 I 


- I - 2 277 


1 ■■11 


jl||p|||l ||H| 






- 1 6 


O 267 


6 Hi 


Bjragroj^K&xixStwSE: 




= 2D 


1 180 10 









I 


- 1 6 1 
1 180 10 
205 
270 4 


1 265 15 

1 257 2 

-1 -1 283 3 

275 15 


6 = 2~D + l 
1 180 


2 :Bk| 


[it < 1 










m jl 










ailifififll 










|H9 











7 6 



Mr. Ernest W. Brown, Theory of the 



366. All 'planets. 
S V= - o // -02 4 sin (V - 5°) - o // -oo 4 sin (l'±l - 5°) - o"-oo2 sin (2' + 1 - 2D - 5 ). 

Latitude. 
367. <$£/= +o"-ooiCsin{0+ i ; , T + i(T-V) + a o } = + o"-ooi Csin ^, Fcmwj. 



= ±F 



J 


?. 


a 





I 


O 




2 


180 


I 


2 


2 73 




3 


272 


2 


3 


201 



?= -F + 2D 



a 

4 

7 
6 

2 
2 



2 





7 


I 


180 


M 


I 





16 


2 


180 


23 


3 




2 


2 


271 


9 


3 


272 


6 


3 


268 


2 


2 


269 


6 


1 


101 


1 


2 


280 


1 


3 


199 


3 


3 


34i 


5 





(9 = 


= F 


+ 2D 


i 


'' 




a 





- 1 




180 




1 









2 




180 




3 






1 


2 




271 




3 




272 


1 


- 2 




268 




1 




IOI 




2 




280 


2 


3 




199 


2 


™3 




341 



6 = ± F + / 



c* 

2 

5 
9 
1 
2 
2 
1 
1 
1 
1 
2 



I 





2 


2 


180 


4 


2 


271 


1 


3 


272 


1 


3 


199 


1 


_-_ 


- F + 2D + 1 









f9 = F + 2 D + Z 


i 


i 


E 





I 







2 


180 



(9= ±F + 2 D-J 



o 

- 1 
1 

2 

2 

3 

- 2 



180 
271 

269 
199 



? M-5 T -3 v +2i5°-6 

'A ~5 T + 3^ + 337' 
"'1-2D + 5T-3V+36' 
u\ + l+ 5T-3Y + 216 3 
«W+5T-~3Y + 3 6' 
Wi + ST-sV+^s" 
^-81+ 5V + 67° 



3 

3 
6 

8 
3 

2 
1 

c 
77 
3° 
3 
4 
4 
3 
7 



368. r)Z7= + o"-ooi Csin {d +f'M + i(M - T) + a }, Jfars. 
^=±F 6 = F- 2 D 



J' 


I a 


a 


3" 


i 


a 





2 180 


3 





- 2 
2 



ISO 




= +F + /-2.D 




1 


1 


223 





~ 2 


% 


- 1 


- I 


316 



2 2 



269 



Motion of the Moon. 



77 



369. SU= + o"-ooiC sm{6 +f'J + i(J -T) + a } = +o // -ooiCsin ^, Jupiter. 





6 


= + F 






= 


= F+ 2 D 






0= +F + Z-2D 




j" 


i 


a 


c 


/' 


i 


a 





j" 


i 


a 


a 





I 


180 


9 





I 


180 


8 





- 2 


l80 


4 


I 


O 


48 


8 




2 





3 




- 1 


O 


11 










1 


O 


350 


1 




1 


180 


2 




6 = 


-F + 2D 






I 


237 


2 




2 


O 


15 












2 


352 


2 


1 





357 


2 





- 2 

— 1 


O 


3 
6 


- 1 


O 





2 




2 


171 


6 












2 


273 


2 


- i 


- 1 


302 


2 




1 


180 


29 




3 


102 


1 







186 


2 




2 


O 


12 


















I 





35° 


7 












0= ±F + 


2I — 2D 






1 


237 


6 




= 


--±¥ + 1 







2 


357 


2 




2 


3S 2 


2 





- 2 





2 


1 


2 


354 


4 


I 


- 1 


3°3 


2 




- 1 




6 















181 


4 




1 


180 


8 




* 




C 




2 


273 


3 




2 




1 


Wj, 


+ J 




5 




3 


102 


1 


1 





301 


4 


?,«! 


+ 2J+ 168° 




35 












1 


239 


2 


W t 


- 2J + 24° 




18 




= 


F4-2D 






2 


35° 


1 


«l 


+ 3J + 156 




2 





- 2 





4 


- 1 


_ j 


301 


2 


«'l 


+ I+2J + 168 




2 




- 1 


180 


3 







240 


4 


»1 


-Z+2J + 348 




2 



370. 



Terms acWecZ to <Ae Elements. 



Sw>!= +o" , ooiCsmi/f+s"'82!f 1 , 2 



i 3 T-8V+ 3 i3°-8 
Z+i6T-i8V + 33i° 
Q-4T + 239 
8M-4T + 310 
9 M- 5 T + 3os° 
10M-6T + 306 
HM-6T + 33S 
13M- 7T+ 19 
1SM-8T + 43 
17M-9T + 63 

W s +2 76° - 2 



C 

234 

6* 

3 

3 
8 
2 
6 
6 
26 

4 
289 



8 ( w i - "'2) = + i6"4 + 44"'i4 2 

+ i"T29 sin («« 3 +276°-2) 

8w s = +s\ + 6"-46!! c 2 

+ is"'59 sin («%+ 27 6 °' 2 ) 

Sy = +o" - 698 cos (» 3 + 96°-2) 

S(w'tf + «')=+ !"" 8 9 sin ( r 3T - 8V + 134°) 
+ o"-20 sin (isM - 8T + 216°) 

8w'= 4-0° -32^ 
8e'= - 'oo248e'4 



Mr. Ernest W. Brown, Theory of the 



CHAPTEB XIII. 

ACTION OF THE FIGURES OF THE EARTH AND MOON. 

Section (i). The Disturbing Function for the Figure of the Earth. 

371. Let A, B, C, I be the moments of inertia of the Earth's mass about its three 
principal axes at the centre of mass and about the line joining its centre of mass with 
that of the Moon ; E, M the masses of the Earth and Moon. Then it is well known 
that the disturbing function R is given, with sufficient approximation, by 

R=(E+M) A+ J^ + S ) S3 I , , . . . . (1), 

2r s E 

since the next term of R will have an additional factor of the order 6cr 2 , this being the 
approximate ratio of the square of the radius of the Earth to the distance of the Moon. 
It is true that there is a term with a factor of the order 60 ~ 1 , but this term is exceed- 
ingly minute, owing to the approximate symmetry of the Earth about its principal axes. 
Let V, U, a, S be the longitude on the ecliptic, the latitude, the right ascension 
reckoned from the A-axis, and the declination of the Moon; -^ the precession, e x the 
obliquity of the ecliptic. If P be the pole of the ecliptic, Q that of the Earth's 
equator, the parts of the spherical triangle PQM are : 

PE= h , QM= 90 - 8 , MP= 90 - U, QPM = go° -V-if,, 



and therefore 

Also 
so that 



The second term of this is quite negligible : its principal arguments have daily 
mean motions of the order 3. io 6 seconds, and A - B is known to be very small compared 
with C - |(A + B). Hence 

R = (E+M)'?£(§-sm*8), where «V = JL(o - ^±^"\ . '. . . (4). 



sin 8 = cos €j sin U+aine 1 eos £7 sin (V+ij/) ..... (2). 
I = A cos 2 a cos 2 8 + B sin 2 a cos 2 8 + C sin 8 8 , 
A + B + C-~ 3 I= 3 (c-A±I J Vi-sm 2 8)-f(A-B)cos2acos 2 S. . . . {3). 



Motion of the Moon. 79 

372. Transformation. — -Since 

p cos ( V - T) + i,p sin ( V - T) = u, s , r sin U= z , r cos U = p , 

equation (2) gives 

r sin 3 = z cos e,+ — sin e 1 (we( T +W t - se -(T+W 1 ) , 

21 

1,2 _ f 2 s iu2 S= (|)' ! - z 2 )(i - § sin 2 fl ) + J sin 2 fl (aW+«' + s%- V+»') - 1 sin 2<r 1 (we<T+<M' - ge -(T+«t). 



r Fhe last two terms are the real parts of \ sin^ . w 2 e 2(T+ ' w ' , zi . sin 26! . we (T+,w \ Hence ii? is 
equal to the real part of 



where 



:*{ m-^t ■} ^.(T)'M + ^r#-« 



Mr 



4 /* / 



-s- sin z «, 



u,= -i~sm z e, , ij., = l j sm2e 1 
6 vi 2 in 1 



fi being given in (4). 

The values of the Moon functions have been given in Sect, (v), Chap. X. All the 
terms have the factor p., which is treated as a small quantity of the first order, and we 
should properly put e x = const, and ^ = o. But it is convenient to retain the mean 
motion of ^, as this motion affects the arguments to a slight degree, and is retained with- 
out any increase of labour. The m» are then constants which take the place of m"/m! in 
the equations of variations. In forming j u it is to be remembered that m contains n 2 . 



Section (ii). Numerical Results. 
373. Adopted values of the constants. — I take 

giving 



£] = 23° 27' 32", daily motion of \\r = +o" - 14, /x=+[7'6658] 



M^+tS'925 1 ]. ft=+[4'3 8 36]. ft=+[S'4i9i] .... (8). 

The only one of these constants of which the value is doubtful, within the limits 
of accuracy required here, is fi. It will ultimately be determined by the coefficient of 
the argument w x + ^ in latitude, the principal term arising from the figure of the Earth. 
1 adopt a value here to correspond with that marked (/?) in my paper " On the Degree 
of Accuracy, etc." * ; this is obtained by comparing Hansen's observational value with 
Hill's theoretical values (which closely agree with those obtained by me) for the 
coefficient in question. 

374. Final results. — I omit terms whose coefficients are less than o ,/- oc>3, and 
obtain, for the terms in longitude, 

&V= +o" , 020sin(2D-£) + o"*oo4sin (2F-Z) -o" - 038 sin (2W 1 + 2f- 2F); 



; 



mm 



* Monthly Notices, vol. Ixiv. p. 531 



8o 



Mr, Ernest \Y. Brown, Theory of the 



in latitude, 

8U= + o""o83 sin (210 x + 2f - F) - o"'oc>3 sin (210 x 4- 2f - F - 2D) 

±o"'oo5 sin (2W! + 2i/f-F+Z)- o"-oi7 sin (wj + i/r) - o" , oo7 sin (wj + i/r - 2D); 

and, added to the elements, 

Su>j = + y"-^ 1 7 siu (w g 4- f) , 8w 2 = + 641% - 2"'092 sin (to 8 + f) , Si« 8 = - 600'% + 96"'69 sin (10 3 + f) , 
S» = - o" - oo9 cos (iv s + '/'), Se = + o"'oo2 cos (a> 8 + \j/), Sy = - 4""3 5 1 cos (ie 3 + xj/) , 

of which <$», Se may be neglected. 

The principal term in latitude which results from these values is — 8 / '*355 sin (w 1 + ^ ). 

Section (iii). The Action of the Figure of the Moon. 

375. The Disturbing Function is of the same form as* that for the figure of the 
Earth. Let a' denote the longitude from the A'-axis on the Moon's equator of the 
projection of r on this plane, and S f the inclination of r to the same plane. Then if 
A', B', C ; , V be the moments of inertia, the C'-axis being that perpendicular to the 
Moon's equator, and 1' the moment of inertia about r, we have, as in § 371, 

A' + B' 



A' + B' + C'-3l' = 3 C 



(C' - ^±Z)(! - sin 2 8') + |(B' - A') cos 2a' cos 2 8' . 



(9). 



376. Transformation. — Now the Moon always turns the same face to the Earth, 
and, if we neglect the small real (not apparent) librations, its angular velocity about the 
C ; -axis is therefore constant and equal to n. Moreover, it is well known that its 
equator and the ecliptic intersect in a line whose longitude is w s ; call this point on the 
celestial sphere Q. The mean angular distance of the A'- axis from Q is therefore w 1 — vj s . 
Hence, from the right-angled spherical triangles having each a side, one on the ecliptic 
and the other on the equator, and a common hypotenuse QM, 

cos 8' cos (a +w l - u> 3 ) = cos U cos (V - w a ) . 

If we neglect S\ IT, this gives </= V—w x . Put a' — V—w x + oV; then oV depends 
on squares of S', U, and cos 2d = cos 2( V— w x ) ~ 2S0.' sin ( V— iv t ). As we shall neglect 
quantities of an order higher than the second with respect to the eccentricities and 
inclination, and also the inclination multiplied by mr, we can neglect the second term 
of this last expression. Also if 7 = sin ■§•»', and if —i x be the inclination of the lunar 
equator to the ecliptic (it being well known that the ecliptic lies between the, mean 
lunar orbit and the lunar equator), we have with sufficient accuracy 

siu 8' = sin (i + ?',) sin ( V. - w g ) = sin (i + i t ) sin (w x - w a ) . 

Substituting, we have, for the disturbing function 

i - sin 2 (i + n'l) sin 2 (v^ - w s ) + (/*"///) cos 2 ( V - w,) { 1 - sin 2 (i + %) sin 2 (w t ~w a )} 



R = (E+M) 



0?jX 



where 



2M 



A' + B' 



a 2 fi' 



= -3_(B'-A'). 



Motion of the Moon. 81 

^•jj. Form for computation. — The principal periodic terms have all short periods, 
and we need only consider the constant parts which give small constant additions to 
b.- n b s . Now, if we retain quantities of the orders previously noted, that is, those parts 
which are of the second order with respect to e, y, the portions depending on e 2 will 
alone affect h, and those depending on y 1 will alone affect b 3 . Hence for the former 
we can put 



and for the latter, 



R = (E+M)^[y + n" cos 2(F-w,)]; 



R = (E + M) a f - \ix - !//'] sin 2 (/ + «',) , 



in which terms of the order fm 1 have been neglected. 

Let p e be the coefficient of e l in a s /p 3 , p c that in a?p~ s cos 2 (V— iv-f or in a s p~h(^' 2 . 
Then, referring to the disturbing function for the figure of the Earth, and remembering 
the formulae for obtaining Sb.,, $b 3 (§ 270, Chap. X.), we see that the values of §b 2 , Sb s 
for the figures of the Moon and Earth are respectively in the ratios 

1>-Pe + 3p"p< '■ K J ~ f s i u ' 2 6 i)/Y nn d (l J - + p')-y. s ' n2 ('• + b) '■ l J '( T _ f sln2 6 0^-. s " 12 i ■ 

578, Numerical results. — From the results in Sect, (v), Chap. X., and in the first 
section of this chapter we have 

A- + '386 , flf =--6j8, /,,= +[7-6658], 1 -#sin 2 £ 1 = +7623, S6 2 =+6"-4i, 8l> 3 = - 6"-oo , 

the last two being the values for the figure of the Earth. Hence, for the figure of the 
Moon, the annual mean motions in seconds of arc are 



S&.,= 8"-4i^-44""3— , 8/ ^ ; 

t x /J. 



7 ".g„/* +M sin 2(i + h. 



fl. Sill 21 

f shall now assume that the Earth and Moon are of similar constitution, so that 

C fl : C;E in that ratio of the squares of their diameters, that is, as ('2 jff : 1. I also 

take 

A + n 



lence 



2O 



„/ A' + B" 

T Q E I 1 - 



— = •00328, *'= 5 c "i , ?'i=i L 'S (§379) • 



,/ir - a'\ 



A' 



37c), Adopted values for the mechanical ellipticities. — -The results of Dr. E. 
Hays,* for the lunar librations, give 

i, = i'-32', B'- A'-= +-OOOTS7CA C- A' =+ -0006290' . 
If we accept these values, we obtain for the annual mean motions 

0/)., -- + o"'o3 , So., = — o' ■ 14 . 

In order to obtain these quantities accurately to o"-oi, it is necessary to know the 
two mechanical ellipticities within 5 per cent, of their true values. 

* Abh. der Math.-Phys. El. der K. -Sachs. Gess. der Wiss., vol. xxx. (1907) p. 69. 
HoYAL ASTROX. Soc, Vol. LIX. H 






Nil 



WA 







82 



Mr. Ernest W, Brown, Theory of the 



CHAPTER XIV. 



THE REMAINING PERTURBATIONS. 



Section (i). Corrections due to the Masses of the Earth and Moor 



/2~/3 



380, Correction due to the substitution of m' instead of m' + JE + M for n n a 
This is noted in § 4 (a), Chap. I. It amounts to diminishing the disturbing function 
due to the Sun by the factor i-(E + M)/m f . The method of Sect, (iii), Chap. XII., 
might be used, but it turns out to be troublesome. It is more simple to use the 
ordinary method for the indirect inequalities by putting Sp' = [E + M)/^m\ SV = o for 
periodic terms, and R ~ - (E + M)F Q jm' for the constant term, where jP is the 
constant part of the disturbing function due to the Sun. 

381, Results. — I find, as in an earlier paper,* for the annual mean motions, 

8& 2 = - o"*69 , S/> 3 = + o"'i9 , 

the constant changes of n, e, 7 being insensible in the coefficients of the periodic 
terms. 

The periodic changes are : 

8V= - o" , oo7 sin 2D - o"'O20 sin (2I) - I) + o"'oo3 sin V + o"'ooi sin (1 + I' — 2D) . 

382, Corrections noted in § 2, § 4 (c) of Chap, I. — The former gives §b z = — o" - oi, 
and the latter §b 2 = +o"'02, §h s = — o" - oi. 



Section (ii). The Terms of the Second and Higher Orders, 

383. Sources of the terms. — In the four last chapters we have computed the 
perturbations due to various causes, on the assumption that certain factors which 
multiplied the disturbing functions were so small that their squares could be 
neglected. It remains to examine with some care to what extent this assumption 
is justifiable, and to correct the expressions in the cases where it is not true. 

Let E(r', V, z') be the disturbing function due to solar action, R P that due 
to a planet, R E that due to the figures of the Earth and Moon, and R e that due 
to the motion of the ecliptic. Then if SV, § 2 V, h' be the terms of the second 

* Monthly Notices, vol. lvii. p. 567. 



Motion of the Moon. 



8- 



order in the motion of the Sun, the complete disturbing function for actions from 
all sources, except that in the main problem, is 

m=R{r' + %r' + W, V + 8V+SW, z' + 8z')-R(r', V, o) + %R P + R E + R e . . . (i). 

We have previously neglected quantities of the order (SB)' 2 , and have used elliptic 
expansions for the coordinates of the Sun and planets in the last three terms. 

384. It is convenient to divide the perturbations of the second order into 
classes according to their nature or the sources from which they arise. I denote by 
§ a perturbation of the first order, and by P one of the second order. The several 
portions of S 2 R to be considered are as follows: — 

(A) Terms due to the substitution of c { + Sc ( , w, + hv { instead of c t , Wi in the 
right-hand members of the equations of variations. 



(B) 



(c; 



dr V oz 



-hV . + e .—\Ii. 

oV oz J 



(D) Additions due to periodic perturbations of the solar and planetary coordinates 

in z,lbp. 

(E) Changes in §R due to secular or quasi-secular variations of quantities which 

have been treated as constants. 

(F) Changes due to the secular variations of the solar and planetary arguments. 

(G) Third-order terms due to large second-order terms in the solar and 

planetary coordinates. 

(H) Secular variations of the second order in general. 

It is obvious, in the first place, that the only possibilities we have to consider 
are secular terms in tv x , w 2 , w s , and terms whose primaries are of very long period. 
In the second place, it is to be remembered that the variations of the elements 
contain terms of two kinds — those multiplied by the period of the primary, and 
those multiplied by the square of the period ; and that the latter, for terms of very 
long period, are large compared with the former. Hence, except in the cases of 
arguments independent of the %i\ (in which case the latter terms will be shown to 
disappear), it is sufficient to discuss only these latter terms. Finally, the greatest 
effect of these terms on the coordinates occurs through the change in w u so that 
it is generally sufficient to discuss t^w v 

The various classes are considered in the following paragraphs. 

385. (A) The canonical equations of Sect, (i), Chap. X., will be used and developed 
111 a general manner for the second-order terms. The chief results obtained under 
this heading, namely, a proof that such terms are insensible, practically consists in 
snowing that a given argument arises in two ways, and that, whenever the two 










illil 




i 











1 



8 4 



Mr. Ernest W, Brown, Theory of the 



parts might be separately sensible, they are opposite in sign and nearly equal in 
magnitude. 

Let S^Cf, Shi'i be the additions to c i} w t due to the substitution of c t: + <!c £ , 
ti'i + Swf instead of c i} iv { in the right-hand members. Then the equations for the 
determination of these additions are 

>-$■ ***-- fW£ s 'A^i) !s - ■ ■ ■ » 

Let the first-order variations §c i} hv { be due to a term * in R 

A cos <f> = X cos (pi + a) = A cos (J^'Jj +i 3 «2 +j s w B + f), 
where 4^ is independent of the iv { . Then 

Sc^jf- cos <jf> , 8^=(- ^ - - 1 - — ) sin ^ . , . . . (3). 

Let any other term of R be 

A' cos <jfc' = X' cos (p7 + a') = A' cos (Ji' t i'i+h w 'A + Js' w s^ f') > 

where \|/ is independent of the w*. Let us consider these two terms alone ; then 
the terms in §%, §*w { will have the arguments <p±<p'. 
Put 



Then 



.3.3 . 3 _ 3 . , 3 . , 3 .■ ' 3 _ 3 
Jl 3^ +)2 3^ +Ji dc~ B ~ 3c ' Jl &; +Ja 3iT 2 +Js 3^ ~ 8? 



s 3fi _, /3 2 5 s , 3 2 £ » 

°— =Si(r— „ t,C k + -—^ 3.«»i 



(4). 



.,A dA' . ,, , ,,/AA' dp A' JX\ ,, . , 

= ->; - . — Sill COS <£ -Ji I _ ' - — COS (£ Sill <£. 

p rfc \jp» dc p de/ 

To this must be added the term arising by making §w { , §Ci depend on (p\ and the 
derivatives of R on (p, that is, by changing the accents. 
We thus get 



! 8% < 



A'— f i 4 - it) -j^L d S-- 



de \p' p 



»' 2 ' de 



sin tj> cos <f> + 



' similar term with 
accents changed 



and therefore, since dbi/dc k = db k /dci, 



db t 



dt 



Z,£*-i 



,/iX_,rfAVi d// _ 1 djA ..,/ i dp dp 1 dp' dp 
A dc dc/Kp ' dc H p ' dej \p 2 ' de ' rfeAp' 2 ' de ' dc t . 



sin (</> ± $') , 



Denote the right-hand member of this equation by F t sin ((f) + <p') + F- sin (<p - $'). 

* The symbols ji, j 2 , j s are the same as the %, i lp i 3 of Chap. X. ; the j 1; j 2 , j, i of that chapter are not 
needed here. 



Motion of the Moon. 



85 



Again, 
<3>R 



XSCfil 

,i 2 X' X 



d 2 R 



dc, * \ocfiC-,; GCjCWj. 



, ,, JX'/ X fte 1 dX\ •,-,/,, • -i 4. 

■ cos <f> cos rf> - — I — . — — — . -, ) sin <p sin <p + two similar terms 
dalCf p dCi\p 2 dc p de J 



_ p dcde. 



, X' d 2 k , f dX'/ X dp 1 <iX\ similar 11 , , , ,, 
- + —,. — -, — ± { — — . — - — . — , + i ^ cos (<£ ± th 

■i p dc'dd \dc t \p* dc p dc'J term )J vr r 



which is denoted by G { cos (cp + <p') + G{ cos (cp — (p'). 

Finally, 



'A ^Lj, 



d \ 2 7 XXV x-> • > d 



2A 



XX' d 2 j/ 



cos <f> cos </>' = iZ^ cos (<£ + <£'^ 



vfcj/ pp \^™ J k dc k /\ J -— l k dcj % pp dcdcf, 
Substituting these results in the second three of equations (2), we obtain 

S 2 m> 4 = I (Hi - G{) cos (<f> + <p')dt + I 1 1\ sin (<f> + <f>')dt + two similar terms 



H, - G, 



F, 



sin (<f> + <(>') + 



H { - Gi 



Fi 



j sin (</>-<£') 



(5)- 



p+p (p + p'f) xr ' r ' ' V p - p (p- p'f 

Since we have only to search for terms of very long period, the most important 
terms are those which acquire the squares of the small divisors pzhp', and therefore 
the coefficients F, F' are of chief importance. 

386. (a) If p, p' are independent of the w i} we have S\ = o, S 2 w t = o. 

387. (/3) If ji=j! and p—p', the argument cp — cp' is of long period. In this 

case we have 

dp _ dp dp _ dp' _dp _ dp' 
dCf dc,/ dc do dc dc 

Hence F' contains the factor p —p', and c£%. only acquires the first power of the 
small divisor p —p'. 

In no case, therefore, can the small divisor appear in its second power for terms 
independent of the lunar angles. 

388. (7) Ifji=j{ and p^ —y', the argument cp + <p' is of long period. Then the 
possible combinations from the first-order terms show that one of two things must 
happen : either p, p' are not small, in which case A, X' are so minute that there is no 
possibility of a sensible coefficient ; or p, p' are themselves small. In the latter case, 
the second term of F t is the principal one, and it gives for the coefficient, since p= —p', 



XX' 



1 \fdp\dpj_ 



X dp\f X' dp'\ p 2 

p 2 ' dc)\p' 2 ' dcj(p+~j7j 2 " 



' (P + PV\ P 2 1>" 2 / V'' JdCi \p- 

This is of the order §w t . o'iv t . p 2 /(p+p'f, where Sw it <$'w { are the first-order terms with 
the arguments (p, cp'. If C, C are the coefficients of Sw ( , Sw/ expressed in seconds of 
Me, the order of the new coefficient, also expressed in seconds of arc, is 



- C -V-(-P-)\ 

206265 v +pd 



Ss-i 








; :ffi^ 


III 


'•:♦ 


wm-m 


8 


mm 




iii 




■:>•;.::->■:: 




mm- 




mm 


,¥z3 


mm 




mm. 




mm 




iSSS:' 


: :*$ 


W:-fSW 


■jvfS 


sbss; 


■> 


mm 







:■■:&. 


wm 


II 
11 


Sssi¥: 

mm 

mm 


11 


mm 


Wi 


Siisi 




llli 


:->*•:■ 


mm 




•:•:■:•:■:•:•>: 




:•:•:■:■:■:•:■::■ 


-;^>'^;^;P 




II 
II 


:■-■.■■.•:•:■■•:■. 

mm 






■:■'•'■'": 


: ; : : ' : : : ' : :": : :-:i 




:£:■:•!-:-:-'■:■ 


11 


mm 


lllllli-i 


;ii 


mm 


SK* 


mm 



mm 



'Mmmi 
HIIIII 

mmmm 



mmm 



mwmmi 



86 Mr. Eenest W. Brown, Theory of the 

389. (S) Ifj\=j 1 ' and p±p', the terms in F{" which have the divisor (p—p'f 
have also as a factor one of the derivatives of 6 2 or b 3 to c 2 or c 3 , that is, the factor -to 8 . 
If j9= —p' t the argument in (7) still holds for the principal parts of ^w x , the most- 
important variation. 

390. (e) If j l = o, and p=p', then — F//(p -p'f becomes 



Hf = o, and 



X dX dp 1 . , , , / \ dp\f p 

P «'- *A {P-P? ' "A r dcPXp-p 



(J- 
v -p' 



X dVC d,X 1 cZA.'" 

j? " dcdct ' de,i ' p ' da _ 



1 ^_ j p d I X dX'' 
p-p 2 p~p" de}\p'*" do, 



The principal term is the former of these two. 

Other cases are treated in like manner. The net result is that terms independent 
of the lunar arguments only acquire the small divisor p - p', while those containing 
the lunar arguments are at most of order (Sw^S'w^Kp-p' 2 ). A somewhat closer 
examination is only necessary when the latter class of terms cannot be neglected at 
this first test. No sensible terms have been found. 

391. (£) For secular terms, we have 8c i: = o, dB/dw^o, and therefore § 2 c t = o, 
Shv-i — o. However, certain changes have been made in the arbitrary constants givino- 
changes o c f . But these only produce constant changes in the d^w^dt, and therefore 

* further changes in the arbitraries ; the latter are insensible. 

But secular terms may arise from the combination of two terms of the first order 
of the form (a + bt) sin (p, (a' + b't) sin <p', where (p, <p' are either equal or where their 
periods differ by so small an amount that <f> — <p' may be treated as secular, and 
a, a', b, V are constants. Owing to the minuteness of b, b', such terms are entirely 
insensible. 

392. (B) These terms are treated in exactly the same way as the indirect terms 
of the first order. They are computed in § 402 below. There are no sensible portions 
depending on <V. 

393- (0) These terms are of the order of the indirect disturbing function of 
Chap. XII. multiplied by 3$/ or SV. If p be the mean motion of a term obtained 
in that chapter, and p' that of a term in 3p or V, the worst case is easily seen to be 

of the order 

3.10 p 1 p . 

■■-- r - F - . — — ,= • —i — ; seconds 01 arc , 

200205 p-p 7000 p-p 

for terms independent of the lunar angles, since the largest term in Sp' or SV is less 
than io", and there is no term due to indirect action having a coefficient so great as 2" . 
For terms dependent on the lunar angles we obtain the order 

' ' seconds , 



7000 \p - p 



and a brief examination is necessary. It is easy to see that, if w x is present, op', SV 
must have periods comparable with the month, and must, therefore, have coefficients 



tel* 



Motion of the- Moon. 



17 



too minute to produce sensible effects. When w 1 is not present, there are only one or 
two terms to consider (the most probable of which arise from the action of Jupiter) ; 
these were examined and were found to be quite insensible. 
394. (D) These inequalities cause additions 



>mm 
;'§al 

:' l??X| 

nISI 

III!! 

: -tew**:*:-:! 



sm-- 



8p'(-3-/)+8p"-/+8(F' 






%R P for inner planets . 



(6), 



S 2 i? = 



8p.I+8 P "(- 3 -I) + 8(V'-r") 



A+AY 

de drn'J_ 



%R P for outer planets , 



(7), 



where Sp", SV" refer, in any term, to the same planet as that to which the R P for that 
term belongs : R P is confined to that part independent of 



a a . 



and / is defined in 



is 2 



96, Chap. XL 



To obtain the order of magnitude of the coefficient it will be sufficient to consider 

terms for which / or the derivatives with respect to e', ts' produce a factor not greater 

than 20. Also, since Sp', Sp" are of the same or of higher order of small quantities than 

SV, SV", we use the latter. Proceeding a.s in case (C), we obtain for the maximum 

order of Shv x 

SV-S(V'-V")f p \^8V-8(V'-V")( p 



p-pJ 



-P-P 



iiii 

; : ; : : : : ; : : : : : : : : : : : ; :;' : :v : :: 



111 






ill 



III 



in which every term, as w r ell as the result, is expressed in seconds of arc. 

For inequalities independent of the Moon angles, the factor p/(p ~p') only occurs 
in its first power, and we use Sp' or Sp" in the place of S(V — V"), 

This method provides for terms which arise with a sensible first-order term, that 
is. with a term in SV already obtained. But there may be other combinations due to 
terms in S V previously neglected. Since S(V — V") must be very minute for terms 
of short period relative to the year, we need only consider terms of long period in SV 
relative to the month, and from such terms those containing multiples of u\ higher 
than the first can be certainly excluded. A detailed consideration (which will be 
omitted) of the manner in which the various quantities enter into SV gives a maximum 
value for Sho lt in the case of Venus, of about 2oS(V — V')j=?, where s is the number of 
seconds in the daily motion of the angles <p±(p', and SV, SV" express the number of 
seconds of arc in the coefficients of the terms in V, V" under consideration. 

395. First let us exclude from SV" all perturbations but those produced by the 
Earth. Then the maximum value of S(V — V") is 10", and if we neglect coefficients 
hi S i io,_ less than o^'i, we obtain a maximum value for s of 44". Similar considerations 
for all the planets give numbers exhibited in the following table, which shows the 
maximum value of s for different combinations of SV with R P or R % . In the principal 
diagonal, where the suffixes are the same, S(V - V") must be understood; the 
suffixes denote the sources of the terms. Values of s less than 1" are put down as 
zeros. The last row T refers to the ellipticity terms ; R e is too small to give anything 
sensible. 



lllllli 






MM 



islili 



"■■■>»■» 



Mr. Ernest W. Brown, Theory of the 
Maxima of s for a Coefficient in $V of o"-i. 





sr; 


*f7 


8r V 


« J/ S ' 


»'V 


B, 


44 


44 


24 : 


24 


2 


\R, 


44 


6o 


24 ; 


24 


2 


#M 


1 


7 


? 


4 


O 


A\ 


14 


14 


7 I 


7 


O 


H q 


7 


7 


4 


4 


2 


II. 


44 


44 


7 


14 


/ 



Since the coefficient in <? 2 F varies inversely as the square of s, this table shows 
that for a coefficient of 1" in § 2 V the value of s must be less than 6", or the period 
greater than 600 years. 

The numbers in this table are only rough approximations ; but even if they ought to 
have been twice as great, it would simply have meant that the corresponding coefficient 
in S^V should be taken as o ;/ '4 instead of o"-i. It will in any case retain any sensible 
coefficients. In fact, for periods of over 100 years or so, the minimum sensible coefficient 
will certainly be less than o"-it„ where t„ is the number of centuries in the period. 

The various combinations have been considered in the same manner as that 
employed in Sect, (iii), Chap. XL There appeared to be only one which might, give 
a sensible coefficient, namely, the combination 

(/+i6T-i8V)-(r3T-8V) = / + 3 T-ioV, 

which has a period of 1900 years. This is therefore a second approximation to a term 
due to direct action and given in Chap. XL 

There are no sensible combinations of terms independent of the lunar angles. 

396. In the above, all terms in SV", 3p" which arise from planets other than the 
Earth have been omitted, but for these omitted terms the mutual perturbations in the 
lunar disturbing function cancel one another very nearly. For example, the great in- 
equalities in Jupiter and Saturn with argument 58 - 2J appear in both i? s and E, h Now, 
Leverrier has shown * that the effect of such terms on the motion of the Earth may 
be neglected provided we take J, S to represent the mean longitude, as affected by the 
great inequality instead of the mean longitude alone. But, for such terms, the lunar 
disturbing function for the direct action only differs from the Earth's disturbing function 
by certain constant factors and by certain operators k + k 3 a'd/da' + h i (a'd/da'f, 
k , k t , & 2 , . . being certain constants independent of the disturbing functions (Sect. (ii). 
Chap. XL). The terms produced by k nearly cancel one another, while Jupiter 
and Saturn are so far away that the other terms are very small. A rough calculation 
shows that terms from this source are quite insensible. 

* Ann. Obs. Paris, vol. ii. 



HP 
III 



1 



SB 

mi 

11 

i 

IP 



Motion of the Moon. 89 

Hence, the terms of this nature are sufficiently accounted for by adding the 
" planetary perturbations to the mean motions of the arguments of the planets. 

397. (E) For these secular terms, the investigation of Sect, (iii), Chap. XII., shows 
that it is sufficient to insert the disturbed values in the final results. The only term 
which can be affected is the great inequality due to Venus. This depends chiefly on 
y"°- and tyj"/' = tcl ZA-OO, where t e is the number of centuries from 1850-0. The 

: , maximum change is therefore less than o"'oi per century and is quite inappreciable. 

398. (F) These motions should properly be inserted in the arguments before 
' .division by s or s 2 , and this is sufficient to account for them. There are only two 

terms which can be sensibly affected, those with arguments Z+16T-18V, and 
■/ + 3T-10V. The term with argument 2^ + 2tar', period 10,000 years, due to the 
figure of the Earth, has a coefficient less than o" - oi (according to Dr Hill,* o' ;, oo2 5). 

399. (G) There are a few second-order terms in the motions of the planets which, 
when inserted in the disturbing function for the direct action, might produce sensible 

"■ third-order terms in the Moon's motion. A list of these was made and the possible 
• combinations considered, but nothing sensible was found. 

400. (II) Secular variations.— It is a fact well known in the planetary theory 
that the secular variations of the planetary elements do not produce secular variations 
in the function which is the inverse distance between the planets. But the disturbing 
function which has-been used for the direct action depends mainly on derivatives of 
the planetary disturbing function with respect to the Earth's distance and mean 
longitude. Hence the secular motions of the planetary perihelia and nodes can 
produce no secular changes in the Moon's motions through the direct action. In the 
same way the part arising through fa! in the indirect action is insensible. 

We have then the ellipticity terms to consider. Here we have taken e x , the 
inclination of the ecliptic (which was considered as fixed) to the equator, to be constant. 
...When the motion is referred to the moving plane of reference, it might be thought 
that this would introduce a secular term of the second order. But the principal part 
hi the motion of the ecliptic only produces a periodic term of period equal to that of 
; the Moon s node, and the principal perturbation produced by the figure of the 
Ivtitb is also a periodic term whose period is that of the Moon's node plus the 
pivce.sMon. The term of the second order which might be sensible is therefore 
one having a period which is the difference between these periods, that is, a term 
.independent of the lunar angles and of period equal to that of the precession; it 
1* therefore quasi-secular. . But it is easier to treat it by the method of § 385, and 
then the theorem that the first power of its period will be the only multiplier 
fcmdmg to make the coefficient large comes into play. If we expand the term in 
wie form a + bt e + ct* + . . , the portion a + bt e is only a slight alteration to the mean 
{''Ugilude (an observed quantity), and the secular part ct c " will therefore have the first 
l-"j\\cr c the period as a divisor, and consequently may be expected to be very small. 



* Amer. Eph. Papers, vol. iii. p. 342 ; or Coll. Works, vol. ii. p. 318, 
IioVAL ASTRON. SOC, VoL. LIX. 



1 9 



m 

§ 

m 



^ ■ 



9° Mr. Ernest W. Brown, Theory of the 

The period is somewhat altered owing to the motion of t, but the argument is unaffected, 
since t is only present with w s . 

I have computed this term in the following section, and have found o' /- i5 for the 
coefficient of the periodic term, and consequently, for the secular acceleration durino- 
historic times, o' /- oooi^ 2 , a quantity quite insensible. 

The theorem that periodic terms of the first and second orders independent of the 
lunar angles can never receive multipliers higher than the first power of the long period 
practically enables one to reject any possibilities of secular or quasi-secular terms arising 
from perturbations of the second order, whatever may be the source. 

401. Summary. — -We have to find : 

Case (B). The indirect effect of the solar terms whose arguments are 4M — 7T + 3 V, 
3J — 8M + 4T. In longitude these are, according to Newcomb,* 

&V = + "-266 sin(332°"3 + ng°-ot,)+ 6"-4osin (22i c, i + 20°-2t). 

where t c is the number of centuries from 1850-0. 

Case (D). The term of argument Z + 3T- 10V due to substitution of the periodic 
variations of the coordinates of the Earth and Venus of period 13T — 8V in the dis- 
turbing function for the direct action. 

Case (F), The insertion of the motions of the perihelia and nodes in the arguments 
Z+16T— 18Y, Z + 3T-10V, with the resulting changes in the divisors s 2 and the; 
coefficients of the terms. ; 

Section (iii). Calculation of the Terms. J 

402. Case (B). We require §*p', and we only know o 2 F. Now, 8*p' depends maiidv 
on <rV/V, and for terms of very long period we have 

"J a " m 

so that, approximately, 

But the methods of Sect, (i), Chap. X., and of Sect, (i), Chap. XII., give 

approximately; here the suffix o denotes the values of M 1; if 2 from Sect, (v), Chap. X. 
corresponding to 6 = 0. 

This formula, similar to that obtained for first-order terms by Radact,! who,! 
however, neglects M 2 , gives for the two terms mentioned in the previous section 

SVj, = + o"-04 sin (152° + 1 ig°-of,) + o"'84 sin (4i°'i + 2o°-2f,) . 
* Tables of the Sun, p. 19. f L.c, § 279, p. 39. 



Motion of the Moon, 



91 



r -i j 1 ■ 

IBBl 



• o'. Case (D). The term of argument 1 + 16T- 18V+ i5i°T> = <p arises from a 
of this argument in R P . The term of argument I + 3T - 10V + «° = <p - ft arises 



fiom a term 111 



ifR. 



(f^ T ^ 



•here «'T is the term of argument ft in the Earth's mean longitude, and SV a term of 
rh<> same tii'ii'ument in the mean longitude of Venus. 

From Niavcomb's computations* SV — — (1*92/1 '44)iST, so that 

C T_SV= + 3 -i 6 i"-89sm(i3T-8V + i34°)= +4""4i sm (13T - 8V+ 134°) . 
1 '44 

T«) deduce the new term from that with argument cp which has a coefficient in SV of 
•14"-;. we must multiply by - 16 to account for d/dJ + d/dv,', by (13-01/1 -85)* for the 
tJiatisre iu the -value of s' 2 , by ST - SY, by \ for the change of the product of two smes 
tie cosine of the difference of dp, ft, and divide by 206265 for the reduction to 
seconds. This gives 

= -o"-i2sin(Z + 3T-ioV+i7°). 

Tlii- rtdu.-es the term of this period (see § 316) to + o ;/l 2 3 sin (I + 3T - 10V + 2 1°). 

404. Ca„e (F)., The principal part of the term with argument 1+ 16T- 18V+ i5i° - o 
" mod iVom 1+ 16T- 18V + 2I1" ', where h" is the longitude of the node of Yenus. 
ode has a daily motion of -o"-05, so that the daily motion of the argument is 
ii instead of - i3"'Oi. The coefficient (i^'^o,) must therefore be diminished 
jviio of the squares of these two numbers, that is. by o w- 2 2. This is the only 
:• effect that terms of the second order have on the great Venus inequality, 
u- "oeffjeient of the other term depends chiefly on 1 + %T~ 1 oV + rar' + 6/&". The 
x.Hon of the two latter terms of this argument, which were the parts previously 
••<;,. is -o"-26, and of the former three, + i"'85. Hence the coefficient must be 
"1 i:i the ratio ( 1*85/1 -59) 2 . It was found in case (D) to be o /; -2 3 ; its final 
■• rhuvfore +o' /- 3i. 

'-" no, ion of -r in the argument w s — r due to the motion of the ecliptic was 
■d. It has a daily mean motion of +o // '09 J so that the coefficients must be 
•l-'Ui by about 1/1800 of their value. 

'f.. J special term in the secular acceleration due to the motion of the ecliptic 
1 ?hr,',\- of the Earth.- — The arguments of the principal terms due to the motion 
<••hj_.de and the figure of the Earth are respectively (p- w s + g6°-2 -o"'Ggt, 
■r--' m \dl, where the coefficients of t are the daily motions of t and of the 
ion respectively. The argument <p-ft is therefore -~o' /, 23£ + 96° - 2. This is 
ecnlar, but it is more easily treated as periodic by the formulas of case (A) of the 

* Am. JSph. Papers, vol. iii. pp. 476, 488. 



||p|||H|ii 

liiiSlii 



HH! 



mmmk 



lillS 



'~^^^*****la*''l**mmmmmmmmmmmmmmmmmmnmm 



92 Mr. Ernest W. Brown, Theory of the 

previous section. It is independent of the w L , and therefore only receives the first 
power of the period as a large multiplier. 

We have j\ =j 2 =j\' =>/ = o, j a =j 3 = 1 , p = b 3 - o ;/ -09, p' = b 3 + o"- 1 4 . Since b s 
has a daily motion of- 190" "8, we can put p=p r = b 3 , except when they occur in the 
combination p—p f . 

Substituting in the formulae of § 385, and remembering that we only need §hv u 

-F{ _ x 1 ( y dX -,d\'\ 1 db 3 _ XX' 1 _ db. 3 _ db s _ 

— — } - . .1 A -■ + A - - , ' ' ' 



(p-p') 2 ' P~p\ de 3 dc 3 Jbg 2 dc l p-p bJ dc-^ de a 

Now A, X' contain the factor k, and c s = - 2k 2 na 2 approximately. Hence 



~ 1 Y V _ XX' 1 db t 



b., d<\ 



(p-p') 2 2b 3 '(p-~p')c, i di\ 
The second term in the square bracket is of order k 2 compared with the first, and may 
therefore be neglected. 

Treating (x/ in the same manner, and neglecting the terms factored by db 3 jdc 3 , 
which are of order k 2 compared with the others, we find, after some reductions, 

p-p " c 3 b 3 (p-p)\ dcj dcj/ c 3 b 3 (p-p) de 1 



Also, H x is of order k 2 compared with these terms, Hence 

sin (<f> - cj>') , 



XX' db„ d ,,,,.. 

— . — s + — (XX ) 

h dc i dc i 



1 2 (p-v) h c i 
Now, by Sect, (ii), Chap. X., approximately, ! i 

d 1 d n db , „ i 

= _ /3='33. ^=+-0037, & 3 =--oo 4 o, : ; 

dc t a~p an dit i 

Also, X = ka 2 nM, A' = k«V, where m, m' are independent of the c t for the terms of:; 
lowest order, so that AA ; = IC'ahv'p.p.' = c 3 n 2 pi", where //' is defined like p., m'. Hence i 

mv, = l ■ JP-, j^^lz - 6] sin (* - 4>') ■ \ 

But, approximately, | 

- 4na 2 y8y = 8c 8 = — - cos tj> , - qncpyh'y = — cos <f>' ; | 

■> ■' , 1 

and $y= + o"-jo cos <£ (§ 370), S'y = -4 //- 35 cos <£' (§ 374), p = b 3 = - i90 /; '8, 
p—p'=—o"'2T,, (p — cp' = g6''2—o /f '2^t. Hence 1 

SVj, = - 12 ■ i2 — ■ 1 7 A4 — 35.; g ; n (06° '2 -o" , 2*f) = o"-i< sin(o6°-2 - 2°-%t c ) , I. 

- '23 206265 I 

which is equivalent to a modern secular acceleration of o //- oooi^ 2 , and therefore entirely 
insensible. 1 

406. It might still be thought that terms of the second order in the disturbing 
function of Sect, (iv), Chap. XII., will give rise to sensible secular terms. But 9 S is 01 
the third order, and therefore any terms which arise will either depend on the tv { or 
will be constants. It is also to be remembered that arguments in x, y containing odd 



Motion of the Moon-. 



93 



multiples of w s , and arguments in z containing even multiples of w s , only arise through 
the disturbing functions SR, S Z R, and much the largest of the combinations with a long 
' period and of the second order is that just computed. 

Section (iv). Perturbations with Unknown Constants. 

407- The attractions of the minor planets.— No one of these is large enough to 
produce any sensible effect on the motion of the Moon, but their aggregate mass may 
possibly be comparable with the mass of the Earth. The chief force -would be that of a 
riu°" of matter of diameter between two and three times that of the Earth's orbit, and 
the principal effect would be constant additions to the mean motions of the perigee and 
node of the Moon's orbit. But on any supposition which would fit in with the small 
differences between theory and observation for the motions of the planets, these constant 
addition,-: cannot exceed two or three hundredths of a second of arc in the annual mean 
motions of the Moon's node and perigee. 

Periodic effects would be smaller, and the chief of them would have a period of 
one vear, with a coefficient less than one-thousandth of a second of arc. 

408. Other matter in the solar system.— There is undoubtedly a large number of 
meteoric bodies distributed through the solar system and revolving mainly round the 
Sun. The most reasonable supposition is that this matter may be considered as 
arranged in rings of varying density round the Sun as centre, so that the effect would 

■ be that of a thin plate with its centre at the Sun, and of density increasing towards the 
centre. If this density varied as the qth power of the distance from the Sun, where q 
h some negative number, the effect would again be to add to the apparent mass of the 
Bun, to add something to the mean motions of the perigee and node, and to produce 
additions to the known periodic terms. The effect on the Earth would be of a similar 
nature. No secular terms arise. Professor Kewcomb has discussed such hypotheses.* 
In auv case, the effect on the Moon can be neglected. 

409. The action of the tides.— First, neglecting friction, the action of the lunar 
tide chiefly produces a standing wave with reference to the line joining the Earth and 

• Moon. Its effect is therefore similar to that of the figure of the Moon, and there can 
be little doubt that it is quite negligible. The action of the solar tide must chiefly 

..produce a term depending on the difference of longitudes of the Sun and Moon, and is 
similarly too small to be considered. The reaction of tidal friction produces. a real 
secular retardation of the Moon's motion, as well as the apparent acceleration due to 
the slowing down of the Earth's rotation. The former is nearly equal to the latter, and 
•t«e real retardation would be between two and three times the observed acceleration. 
I lier<- being no data on which to base any exact numerical estimates of either of these 
•'{Uaiitititb, the secular acceleration will be considered as an observed quantity, the 
magnitude of the apparent value being not very different from that (5"'8) found from 
the attractions of the planets. 

* Astronomical Constants, chap. vi. 



94 Mr, Ernest W. Brown, Theory of the 



CHAPTER XV. 

THK FINAL EXPRESSION FOE THE MOON'S COORDINATES. 

410. In this concluding chapter I gather together all the perturbations which have 
been found in detail in Chaps. X.-XIV., so that the expressions for the coordinates of 
the Moon in terms of the time are obtained by adding the results given below to those 
at the end of Chap. IX. 

411. The values of the mean motions of the perigee and node are collected in the 
following scheme with the references. 

Annual mean motion of the Perigee, Node. 

Principal solar action (§ 195) 
Mass of the Earth (§§ 381, 382) 
Direct planetary action (§ 316) 
Indirect „ ,, (§ 370) 

Figure of the Earth (§ 374) 
„ ,, Moon (§ 379) 
Final values (epoch 1850-0) 

[The small differences from the values given in 1904* are chiefly due to the some-: 
what doubtful parts depending on the figure of the Moon ; these were neglected in the:: 
earlier paper. The value of the ellipticity of the Earth adopted here is that correspond-:; 
ing to the result marked (/3) in the paper.] I 

412. Notation. — I now use w v w 2 , iv s to represent the mean longitude and 
mean longitudes of the perigee and node with the motions just given ; w, the mi u) 
longitude of the perihelion of the Earth's orbit at epoch; Q, V, T, M, J, S, the me: 11 
longitudes of the planets. These will receive additions in § 413 below, denoted .1)* 
the symbol S. Thus 

D = w 1 4-8?9 1 -T-8T, l = w 1 + Sw 1 -w 2 -Sw 2 , F = w 1 + Sw 1 -w s -Bw a , I' = T + ST - vf - Bm'. f 

The constants y, e' also receive additions §y, <V, given below-. The changes in 
the Moon's coordinates are accounted for if we multiply the terms containing the 
arguments iF, il' by the variable factors 



+ 146426-92 


- 69672*04 


•68 


+ '19 


+ 2-69 


- 1 '42 


•16 


+ - °5 


+ 6*41 


- 6 '00 


+ -03 


•14 


+ 146435-21 


-69679-36 



rs 



, . . 1 Sy 1 . Se 

I + . I i -i • - -—r- , I +;*,—■ , 

y 206265 e 

' Monthly Notices, vol. lxiv. p. 532. 



Motion of the Moon. 



95 



pe>'.th>-lv. A.11 other variations of the constants present in the coefficients are 
be 1 ' m-cnsible or have been included in the expressions for the coordinates. 

The number of centuries from i850'o is represented by t e . 

The arrangement of the tables is the same as that of Chaps. XL, XII. 

All final coefficients below o' /- oo3 have been dropped. 

A star instead of a number in the last place denotes that the last figure has not 
l .ji computed. 







mmm 



96 



Mr, Ernest W. Brown, Theory of the 



413. Terms added to the Arguments and to the Constants. 
<5-«, = + 5"-8^, 2 + 2o"'ooi C sin \\> ; 



+ 


c 


* 





i3T-8V + 3 i 3 c -9 


237 


2l) ~l + 2lT - 20V + 2 73°'0 


126 


Q-4T + 239 


3 


2D-/ + 8T- 12V + 303 


33 


8M-4T+310 


3 


2F - 2D + 6T- 5V + 270" 


54 


9M- 5 T + 3 o5° 


8 


3Z-2l) + 2 4 (T-V) 


10 


HM-6T + 335" 


6 


D+ 12T- 15V+262 


13 


13M-7T+19 


6 


D + 25T- 23V+ 190 


13 


i S M-8T + 43* 


26 


F + 24T- 23V + 285 


3 


17M-9T + 63 


4 


D+Z-F+I7T-I8V+75" 


8 


ii9° - oif ( .+ 152° 


4* 


2D-Z+ST-4Q + U3 


3 


20°'2t c + 4l° m I 


84* 


2D-Z + T- 3 Q + io5° 


75 


/4-3T- ioV-2°-6f c + 3 3° 


31* 


2F-Z4-3T-4Q + 67 


3 


Z+16T-18V- i°-ot+isi°-o 


1427* 


4D-3Z+25M-23T + 67 


4* 


1+ 29T - 26V + II2°'0 


108 


D-F+2M4-16S 


17 


Z+2i(T-V) 


30 


n' s + 2 76 0, o 


282 






H' 3 + l°'4t c 


7317 



8w 2 = -38"-3Z c 2 -o"-ii8sin(Z+i6T- i8V+isi°-o) 

4-o"'84o sin (w s + 2 76 0, 2) - 2"'092 sin (w 3 + i°'4t c ) 

+ the ten periodic terms in Sw t whose angles are independent of w u iv 2 , w s ; 

s "'s= +6"'sZ, 2 + o"'i72 sin (<?+ 16T- 18V + i5i°-o)+ i"-86 sin (w 3 + 29o°-i) 
+ i5"-58sm(t» s -o°-9Z c + 276°-2) + 96"-69sin (» 3 + i 0, 4^); 

ST= + i"-89 sin (13T- 8V+ i34°) + o-" 2 o sin (15M - 8T+ 216 ) 
- 6"'40 sin (2o°-2Z c + 4i°) -o"'2 7 sin (ii9°-o^4- 152°); 

8sr' = o° , 323i ( ., 

8J= +o°-33sin(38°'54+n5°). 3S = - o°-8 3 sin (38°- 5^+ 115°); 

8y= - o" - o83 00s (?» 3 + 290°" 1) - o" - 698 cos (w> 3 - o°'9Z c + 2 76° '2) 
-4" - 35 x cos(m' 3 4-i 0, 4Q; 

8e' = - - 00248e%. . 



■ 



Motion of the Moon. 



m 
1 

m 

1 



P 
■ 



i 



414. True longitude = 2o /; '*oor C sin {0 +/T + i(T - V) + a°}. 



HI 



# = 



179 


3° 7 


3S9'3 


42 





46 





33 





24 



272-9 
271-7 



2D 



34° 

176 



4 
so 



199*0 


92 


204 


26 


1 7 


9 


207 


4 


114 


26 



> = 2 D 



= Z 



*' 


a 





3 


* 


a . ' ■ 





6 


O 


II 





_ 7 


180 


4 


5 


O 


II 




- 6 


180 


5 


4 


O 


8 




- S 


180 


6 


3 


180 


34 




- 4 


180 


6 


2 





3 6 




~~ 3 


180 


8 


1 


l8o 


23 




- 2 


O'O 


61 


1 


- 


99 




- 1 


180*0 


129 


2 


1 79*5 


136 




1 


o - o 


152 


3 


178 


1 3 




2 


iSo'o 


48 


4 


180 


4 




3 


180*0 


127 


18 





3 




4 


180 


11 


1 


2^2 


3 


1 


- 2 


258 


4 


2 


271-0 


40 




1 


75 


8 


3 


271-5 


37 




2 


271 


46 


4 


89 


\ 




3 


272 


40 


20 


2 73 


3 




4 


272 


5 


II 


78 


3 




S 


92 


4 


10 


78 


4 




3 3 


2 t 2- 


6 


9 


78 


4 


- 1 


.- 3 


268 


3* 


8 


7.0 


4 


. 


- 2 


264 


46 


7 


78 


S 




- 1 


104 


9 


6 


78 


s 




1 


102 


3 


5 


84 


7 




2 


282 


7 


4 


78 


7 




3 


280 


7 


2 


271 


19 


2 


-18 


209 


3 


1 


98 


9 




3 


210 


14 


2 


281 


13 




4 


205 


4 


3 


281 


3 




5 


J 9 


3 


3 


199 


11 




6 


198 


16 


4 


202 


3 


2 


- S 


161 


3 


s 


SO 


4 




- 4 


33 6 


4 


6 


162 


6 




- 3 


33i 


*s 


3 


342 


"5 


3 


S 


"5 


4 


2 ■ 


7 


3 


"3, 


- s 


65 


4 


15 


151 


4 










18 


151 


10 











Stlfe 



?RON, SOC, V 



LIX, 



13 



9 8 



Mr. Ernest W, Brown, Theory of the 



True longitude = 2o"'oo i C sin {0 +j"T + /(T - V) + a} . 



0^ 


2 D-/ 






e 


= 2 D-Z 






<9 =22 -2D 




J '<■ 


a 





J 


i 


a 


C 


3 


i a 


c 


O - IO 


180 


3 




-3 


340 


24 





4 


3 


- 9 


i So 


6 




— 2 


6 


3 


1 


4 92 


4 


- 8 


180 


8 




x s 


151 


25 


_ 1 


-3 268 


3 


- 7 


180 


15 


3 


s 


114 


4 




-2 268 


3 


_ 6 


180 


22 










2 


6 17-4 


62 


- 5 


180 


3« 




6 


= 2 T> + 1 










- 4 


i8o'o 


83 





-5 





3 




i9=2l~4D 




- 3 


i8o'o 


658 




-3 





4 





3 


8 


.... 2 


CO 


137 




- 2 





4 








_ J 


180 


J 3 




1 





11 




6 = 3/- 2D 




I 


O'O 


i33 




2 


180 


iS 





3 180 


3 


2 


179-6 


iS7 




3 


180 


3 








3 


178 


14 


1 


2 


271 


5 




6= -4D 




4 


180 


3 




3 


27 1 


4 





3 


7 


18 

I ~~ 2 




259 


1 1 
3 


- 1 


- 2 


269 


3 




(9= -D 




— I 
2 

3 


270 

271-4 
271-9 


3 
65 
49 





8~- 

1 


= 4D - Z 

180 


6 


1 


3 273 
6 = 7^D 


5 


4 


90 


5 




2 





7 


1 


3 273 


11 


i - 8 
- 7 


269 

269 


3 
S 






e=2i 






= w n 




- 6 


269 


8 





— 2 





S 


2 


3 216 


19 


- 5 


269 


24 




- 1 


180 


10 


— 2 


- s 255 - 


3 


- 4 


89 


3° 




1 





10 




~~4 255 


9 


- 2 


268 


38 




2 


180 


6 




~3 75 


16 


I 


IOI 


9 




3 


1 8c 


9 




- 2 75 * 


5 


2 


281 


1.3 


1 


2 


273 


3 








1/ 


2 S3 


3 


- 1 


- 2 


267 


3 




(9 = w 3 + 2F 




2 3 


200 


19 










2 


3 216 


4' 


4 


202 


4 




<9=2?-~ 2 D 




- 2 


-3 4o 


4; 


5 


19 


4 





- 2 





7 








2 - 7 


340 


3 




- I 


180 


S 




6 = ?» 3 ± I 




- 6 


162*6 


79 




2 


180 


3 


2 


5 216 


3: 


^ 5 


165 


4 




3 


i8o - o 


73 









Motion of the Moon. 



99 



415, True longitude = 2o"-coi C sin {0 +j"M + i(M - T) + a }. 



i 


a 


C 


I 


180 


11 


2 


l8o'2 


195 


5 


3S7 


14 


4 


349 


s 


3 


260 


6 


,-i 


224*4 


3 2 7 


2 


2I2'4 


38 


3 


212-5 


48 


.4 


33 1 


10 


; 


244-8 


93 


s 


245 


20 


4 


244 


14 


: -¥ 


62 


6 


? 


2?7 


16 


?4 


276 


13 


5 


275 


6 


6 


94 

#=2D 


3 


•:2.. 





S 


I 


180 


4 


P' 


181 


44 


3 





s 


;'i 


224 


23 


;;2- 


212 


6 


;;3 : 


214 


8 


4 


37 


3 


W: 


149 


3 


|:S 


328 


3 


1 


3 1 ? 


23 




280 


3 


2 


244 


5 


:••! ' 


244 


4 


;:4'- 


246 


4 


- 


297 


8 



f? 


* 


a 


C 





~3 


180 


3 




— 2 





38 




- 1 





4 




1 


180 


5 




2 


180 


43 




3 





3 




4 


180 


3 


1 


1 


223*3 


73 




2 


212 


10 




3 


213 


13 




5 


210 


9 


1 


-3 


33° 


9 




- 2 


S 2 ? 


8 




_ 1 


306-3 


74 


2 


2 


245 


17 




3 


245 


S 




4 


244 


3 




6 


63 


6 


2 


-4 


296 


3 




-3 


295 


5 




- 2 


295 


18 


3 


3 


277 


3 




4 


276 


3 


3 


-4 


264 


3 



)=2~D-l 



5 


180 


3 


4 


182 


20 


3 





5 


2 





1 3 


1 





3 


1 


180 


8 


2 


iSro 


61 


3 


3S3 • 


5 



-■2l)~~l 



i 


a 





I 


220 


31 


2 


212 


II 


3 


214 


14 


4 


27 


3 


-6 


149 


3 


-5 


I S I 


43 


~4 


329 


3 


-3 


327 


3 


- 2 


328 


6 


- 1 


320 


35 


3 


280 


4 


2 


244 


11 


3 


Z44 


6 


4 


245 


S 


~6 


298 


33 


-3 


296 


3 


- 2 


297 


14 






2 


1 


1 


1 


- 1 



^= 2 D4-^ 
180 
82 



93 



0=2l 






- 2 


O 




2 


180 


I 


I 


232 


I 


- I 


308 



:2/~2D 

o 

209 

244 



4 
17 
18 



■ iim 



mm 

1 



IOO 



Mr. Eknest W. Brown, Theory of the 



4 1 6, True longitude = 2o" -oo i C sin { & +j"J + i(J - T) + a' } , 



2.D- l 



'i 


a 





I 


178-8 


643 


2 


3S9 - 6 


187 


3 


7 


10 


-3 


257 


6 


- 2 


274 


18 


O 


289*9 


87 


I 


241-5 


.165 


2 


352-0 


5 2 


3 


355 


4 


- i 


250 


10 


o 


324 


5 


i 


238 


2 5 


2 


344 


6 


I 


230 

#= 2 D 


3 


"3 





4 


— 2 


i8o'o 


70 


- I 


1 


33 


I 


I78-5 


167 


2 


359' 2 


85 


3 


13 


7 


o 


349 


27 


i 


237 


35 


2 


35 2 


x 5 


- 2 


8 


30 


- I 


303 


6 


O 


184 


33 


2 


273 


9 


3 


102 


6 


i 


236 


5 


2 


345 


3 


O 


200 


3 


I 


no 


6 



i 


a 





/' 


i 


a 





- 2 


180 


36 


- 1 


- 1 


296 


18 


- I 


I'O 


144 







I74-2 


60 


I 


I79-0 


158 


1 


2 


273 


16 


2 


iSo'o 


190 




3 


I02 


7 


3 


21 


5 


1 2 


1 


237 


6 


2 


274 


6 




2 


344 


3 





282-3 


62 


: " 2 


- 2 


19 


5 


1 


242 


39 




_ ! 


291 


3 


2 


352-5 


96 










2 


188 


7 




= 


= 2l)+l 




1 


298 


35 





- 2 


180 


3 





257-2 


63 




- I 


I 


5 


2 


273 


6 




I 


178 


21 


3 


286 


8 




2 


359 


7 





326 


7 


1 





353 


4 


1 


238 


5 




I 


237 


5 


2 


343 


4 




2 


35 2 


3 


1 


302 


5 


- 1 





182 


4: 





2.14 


7 




2 


2 73 


i 


= 


2D - / 






= 4l)-Z 




4 


180 


4 





— 2 


180 


r\ 


3 


182 


22 




1 


358 


9 ;: 


2 


180-3 


"37 




2 


179 , 


5j 


1 


1 


5' 


- I 


- 2 


7 


3:: 


1 


178-4 


211 










2 


359-2 


8 9 




#=2l 




3 


14 


6 





— 2 


180 


3 i 


3 


261 


5 




— I 


2 


II 1 


2 


310 


13 




I 


178 


12 1 





5 - S 


56 




2 


180 


io| 


1 


237-0 


46 


1 


O 


2 93 


s| 


2 


35 2 


20 




I 


2 39 


3 


3 


187 


6 


— I 


— I 


301 


3 1 


2 


7'5 


436 




O 


247 


5;| 



Motion of the Moon. 



101 



True longitude = %o"-ooiC sin {e+j"J + i(J -T) + a}. 



2I-2D 



I 



.« 





180 


s 


2 


II 





3 


179-9 


240 


I72-S 


284 


163 


3 



<?= 2 Z-4D 





% a 





2 


I 


2 173 




# = 3?-2D 





2 l8o 


I 


2 172 



c 
9 

7 



o 

o 



45 
168 



417. True longitude = 2o"-ooi C sin {0 +/'S + i'(S - T) + a'}. 



. a 





179-6 


42 





8 


273 


21 


257 . 


13 


297 


3 


D 


c 


180 


10 





5 


270 


4 


257 


3 


25s 


4 



0=z 



i 


a 


c 


- I 





6 


1 


l80 


10 


2 


180 


3 





263 


12 


I 


257 


3 


— 1 


283 


3 





277 


12 



:2D-l 



i 


a 


c 


— 2 


l80 


19 


I 


180 


14 


2 


O 


4 





271 


6 


I 


257 


3 


— 2 


271 


5 





267 


6 






:¥?*S3 



:. 



4 1 8. True longitude — 2o"-oo 1 C sin 6. 



2D 

2D -I 
2F-Z 

r + 180 






9 


c 


10 


2l'+222>'' 


4 


39 


I + 1' + 180° 


6 


4 


I -I' +180° 


6 


35 


2W 3 +2°-8t+i8o° 


38 



s : ss;i 



102 



Mr. Ernest W, Bbown, Theory of the 



4 1 9. Latitude = 2o" -oo 1 C sin ifi +JT + i(T - V) + « c 



= ± F 



-7 
-6 

~5 
-4 

-3 

- 2 
- 1 
1 
2 
2 
3 

•s 

■ 2 

3 

•3 



1 o 

2 180 
2 2 73 

(9 = F + 2l) 

2 O 

1 o 

2 l8o 

?= -F + 2D 

l8o 
ISO 
l8o 
l80 
O 

o 

180 

o 

180 

271 
272 
270*0 
269 

199 
341 



6= ±F + ? 
o 

180 

o 

180 
180 



C 
9 
4 
6 



3 
5 
9 

23 
45 
21 

5 
16 

23 
9 
6 

68 
6 
3 
S 





6 = 


-F + 2D-Z 






i 


a 


c 





-4 


180 


4 




-3 


180 


29 




- 2 





6 




- 1 





5 




1 





6 




2 


180 


8 


I 


2 


271 


3 


2 


-6 


162 


4 





= 


-F+2D-/ 







-4 


180 


4 




-3 


t8o 


3i 




— 2 





6 




— 1 





5 




1 





6 




2 


180 


8 


1 


2 


271 


3 


2 


-6 


162 


4 



i9 = F-2D + ! 
1 S 9° 

= F-2D-Z 

1 5 270 

0= ±F + 2Z-2D 



180 
18 



-6 

-5 
-4 

-3 

- 2 

- 1 
i 
2 
3 
3 

-7 
-6 

"""' 5 
-4 

-3 

- 2 

- 1 
1 
2 
3 
5 

-5 



285 
285 
285 
285 
285 
285 
105 

io 5 
105 

2156 
2 55 
2 55 
2 55 
255 
51-6 

75 

75 
75 
7 5 
75 
125 

67 



3 3 6 

= w 1 + 1 

3 216 
3 75 

3 3 6 
3 2 55 



3 

6 

9 
1.4 



•■' . : 7'>i 



M,,ii,iy, ,,/ the M'iOiK 



[() 



I 



/=F-2l) 



i-- ± V I- / - j L) 



'■ 


j 


; 


ilfliltlsf 


a 


c 







O 


- 2 


fiSlllflllll? 


8 








2 


iSo 


3 






I 


iiiilif 


--3 


5 


mmwa 


llllllll 


I 


- 1 


3 !.6 


"1 



l*S 



1 
.345 



421 . l.'Hihuh- -- ^.i"-mi C -in ','■' + /'"■) + /(.I -T) + >i' j. 



. - I' 



ii I'-:jI> 



1 I80 

o 37 

f.'.- -F-K'P 

3 '- s '' 

! I So 

2 O 

- >5° 
i -37 
c i - 1 

- -7.? 



5 

2<> 
<> 

-'I 
12 

7 
fi 
4 



a. 

iSo 
180 

o 



" 


+ F + / 


1 





1 


100 


2 


i So 


O 


3°' 


-' 


353 





-4° 



422. L<<t'ihi(h ! — ^o""no! (.' sili v'/. 



- 1" - 2 1 • 

?'-! :■ 1S0 

-'■. ■ - T + 7 5' 



42.;. /'.'/■-///,/.,■ ^;. 'voi < ' i-ii< w 



2 1 ) / ,v t v . 1 <o : 

,|»-/-2i.I -T)-i.So" 

.'.I.) ■ / - 3..I H- 2T + 7' 



3 



^ + |.' + /..;]l 



I. so 

O 

O 
17- 



H- il:'+:/-:D 



c 




iliilfiiiiiiiii 


c* 


5 


w i 


+ <r. 


+ 2 -s, 


■\? 


7 


MJ, 


~ w 8 


- _T 


3 


S 


"•1 


+ w 8 


illlilllllli 


5 


7 


•' r i 




- / -i- 1 No 


- 



IbO 

172 



34 
10.3 



C 
3 



c 

4 
1 1 

5' 

20 



7 
35 



ai 



■y-u 

WMKm 



wSjffi 

III 

HI 

m» ■■■ 
ill 



•i 

»:•»:■;; 

■I 

flHI 

flfr 

lis. 

itr 

«■ 

IBE. 

■ 
■ 

IMP™ 

i - : : : : :: :':-: 

fl 

«■■■ 

■■111 

!■■■ 



'■: : &- 



JHBHi 

• .- ■■■.-.• . ; 



1.' l'Niw-.i:-ir-Y, 

1 - - s -1/"'''/ 3. 



FlVIH 



XH